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Iversity  of  California. 

FROM    TIIF.    1  IKRARY    <  >F 

FRANCIS     LIEBER, 

Professor  of  History  jiml  Law- in  Columbia  Colk'jro,  Now  York. 
TJU:  GUT  of 

MICHAEL     REESE, 


CAPTAIN  HENRY  KATER,  V.  PRES :  R.  S . 


an* 


KEVs  DIONYSIITS  LARDNER,L,.IuD. F.R.S.  1L  fcE . 


n   B\'    CLMVEY    6c    I^EA 

Hales  Steam. Pr res. 


THE 


CABINET 


OF 


NATURAL    PHILOSOPHY. 

CONDUCTED    BY   THE 

REV.  D10NYSIUS  LARDNER,  LL.  D,  F.  R.  S.  L.  &  E. 
M.R.I.A.     F.L.S.     F.Z.S.     Hon.  F.C.P.S.  M.   Ast.  S.  &c.  &c. 

ASSISTED    BY 

EMINENT  LITERARY  MEN. 


TREATISE    ON   MECHANICS, 

BY 

CAPT.  HENRY  KATER,  V.  PRES.  R.  S.  &c. 

AND    THE 

REV.  DIONYSIUS  LARDNER,  LL.  D.  F.R.S.  L.  &  E.  &c. 


PHILADELPHIA : 
CAREY  AND  LEA— CHESTNUT  STREET. 

1832. 


llf'J     J/.-H 


TREATISE 


MECHANICS 


BY 


CAPTAIN  HENRY  KATER,  V.  PRES.  R.  S. 

M 


REV.  DIONYSIUS  LARDNER,  LL.  D.  F.  R.  S.  L.  &  E. 


PHILADELPHIA : 
CAREY  &  LEA—  CHESTNUT-STREET. 

1832. 


ADVERTISEMENT. 


THIS  Treatise  being  the  joint  production  of  two  persons, 
it  is  right  to  state  the  portions  of  it  which  are  the  exclusive 
work  of  each.  The  chapter  on  Balances  and  Pendulums, 
the  instruments  on  which  the  measurement  of  weight  and 
time  depends,  has  been  written  by  Captain  Kater.  For  the 
remainder  of  the  volume.  Dr.  Lardner  is  responsible. 


CONTENTS. 


CHAPTER  I. 

PROPERTIES  OF  MATTER. 

Organs  of  Sense. — Sensations. — Properties  or  Qualities.  —  Observation. — 
Comparison  and  Generalization.  —  Particular  and  general  Qualities.  —  Mag- 
nitude. —  Size.  —  Volume.  —  Lines.  —  Surfaces.  —  Edges.  —  Area.  — 
Length.  —  Impenetrability.  —  Apparent  Penetration.  —  Figure.  —  Different 
from  Volume.  —  Atoms.  —  Molecules.  —  Matter  separable.  —  Particles. — 
Force.  —  Cohesion  of  Atoms.  —  Hypothetical  Phrases  unnecessary.  — 
Attraction • Page  1 

CHAPTER  II. 

PROPERTIES  OF  MATTER,  CONTINUED. 

Divisibility.  —  Unlimited  Divisibility.  —  Wollaston's  micrometric  Wire.  — 
Method  of  making  it.  —  Thickness  of  a  Soap  Bubble.  —  Wings  of  Insects.  — 
Gilding  of  Embroidery.  —  Globules  of  the  Blood.  —  Animalcules.  —  Their 
minute  Organization.  — Ultimate  Atoms.  —  Crystals.  — Porosity.  —  Volume. 
—  Density.  —  Quicksilver  passing  through  Pores  of  Wood.  —  Filtration.  — 
Porosity  of  Hydrophane.  —  Compressibility. —  Elasticity.  —  Dilatability.  — 
Heat.  — Contraction  of  Metal  used  to  restore  the  Perpendicular  to  Walls  of 
a  Building. — Impenetrability  of  Air.  —  Compressibility  of  it.  — Elasticity  of 
it.  —  Liquids  not  absolutely  incompressible,  —  Experiments.  —  Elasticity 
of  Fluids.  —  Aeriform  Fluids.  —  Domestic  Fire-Box.  —  Evolution  of  Heat 
by  compressed  Air 7 


CHAPTER  III. 

INERTIA. 

Inertia.  —  Matter  incapable  of  spontaneous  Change.  —  Impediments  to  Motion. 
— Motion  of  the  Solar  System.  —  Law  of  Nature.  —  Spontaneous  Motion.  — 
Immateriality  of  the  thinking  and  willing  Principles.  —  Language  used  to 
express  Inertia  sometimes  faulty.  —  Familiar  Examples  of  Inertia 23 


v  CONTENTS. 

* 

CHAPTER  IV. 

ACTION  AND  REACTION. 

Inertia  in  a  single  Body.  —  Consequences  of  Inertia  in  two  or  more  Bodies.  — 
Examples.  —  Effects  of  Impact.  —  Motion  not  estimated  by  Speed  or  Veloci- 
ty alone.  —  Examples.  —  Rule  for  estimating  the  Quantity  of  Motion.  —  Ac- 
tion and  Reaction.  —  Examples  of.  —  Velocity  of  two  Bodies  after  Impact. 
Magnet  and  Iron.  —  Feather  and  Cannon  Ball  impinging.  —  Newton's 
Laws  of  Motion.  —  Inutility  of. 29 

CHAPTERS. 

COMPOSITION  AND  RESOLUTION  OF  FORCE. 

Motion  and  Pressure.  —  Force.  —  Attraction.  —  Parallelogram  of  Forces. — 
Resultant.  —  Components.  —  Composition  of  Force.  —  Resolution  of  Force. 

—  Illustrative  Experiments.  —  Composition  of  Pressures.  —  Theorems  regu- 
lating Pressures  also  regulate  Motion.  — Examples.  —  Resolution  of  Motion. 

—  Forces  in  Equilibrium.  — Composition  of  Motion  and  Pressure.  — Illustra- 
tions. —  Boat  in  a  Current.  —  Motions  of  Fishes.  —  Flight  of  Birds.  —  Sails 
of  a   Vessel.  —  Tacking.  —  Equestrian    Feats.  —  Absolute    and   relative 
Motion 41 

CHAPTER  VI. 

ATTRACTION. 

Impulse.  —  Mechanical  State  of  Bodies.  —  Absolute  Rest.  —  Uniform  and  rec- 
tilinear Motion.  —  Attractions.  —  Molecular  or  atomic.  — Interstitial  Spaces 
in  Bodies.  —  Repulsion  and  Attraction.  —  Cohesion.  —  In  Solids  and  Fluids. 
Manufacture  of  Shot.  —  Capillary  Attractions.  —  Shortening  of  Rope  by 
Moisture.  —  Suspension  of  Liquids  in  capillary  Tubes.  —  Capillary  Siphon. 

—  Affinity  between  Quicksilver  and  Gold.  —  Examples  of  Affinity.  —  Sul- 
phuric Acid  and  Water.  —  Oxygen  and  Hydrogen.  —  Oxygen  and  Quick- 
silver. —  Magnetism.  —  Electricity  and  Electro-Magnetism.  —  Gravitation. 

—  Its  Law.  —  Examples  of.  —  Depends  on  the  Mass.  —  Attraction  between 
the  Earth  and  detached  Bodies  on  its  Surface.  —  Weight.  —  Gravitation  of 
the  Earth.  — Illustrated  by  Projectiles.  — Plumb-Line. —  Cavendish's  Ex- 
periments.   53 

v  » 

CHAPTER  VII. 

TERRESTRIAL  GRAVITY. 

Phenomena  of  falling  Bodies.  —  Gravity  greater  at  the  Poles  than  Equator.  — 
Heavy  and  light  Bodies  fall  with  equal  Speed  to  the  Earth.  —  Experiment. 
,  —  Increased  Velocity  of  falling  Bodies,  —  Principles  of  uniformly  accelerat- 
ed Motion. —  Relations  between  the  Height?  Time,  and  Velocity.  —  Att- 
wood's  Machine.  —  Retarded  Motion , . .  70 


CONTENTS.  V 

CHAPTER  VI11. 

ON  THE  MOTION  OF  BODIES  ON  INCLINED  PLANES  AND  CURVES. 

Force  perpendicular  to  a  Plane.  —  Oblique  Force.  —  Inclined  Plane.  — Weight 
produces  Pressure  and  Motion. —  Motion  uniformly  accelerated.  —  Space 
moved  through  in  a  given  Time.  —  Increased  Elevation  produces  increased 
Force.  —  Perpendicular  and  horizontal  Plane.  —  Final  Velocity.  —  Motion 
down  a  Curve.  —  Depends  upon  Velocity  and  Curvature.  —  Centrifugal 
Force  —  Circle  of  Curvature.  —  Radius  of  Curvature.  —  Whirling  Table. 
—  Experiments.  —  Solar  System.  —  Examples  of  centrifugal  Force 79 

CHAPTER  IX. 

THE  CENTRE  OP  GRAVITY. 

Terrestrial  Attraction  the  combined  Action  of  parallel  Forces.  —  Single 
equivalent  Force.  —  Examples.  —  Method  of  finding  the  Centre  of  Gravity. 

—  Line  of  Direction.  —  Globe.  —  Oblate  Spheroid.  —  Prolate  Spheroid. — 
Cube.  —  Straight  Wand.  —  Flat  Plate.  —  Triangular  Plate.  —  Centre  of 
Gravity  not  always  within  the  Body.  —  A  Ring.  —  Experiments.  —  Stable,  in- 
stable,  and  neutral  Equilibrium.  —  Motion  and  Position  of  the  Arms  and  Feet. 

—  Effect  of  the  Knee-joint.  —  Positions  of  a  Dancer.  —  Porter  under  a 
Load.  —  Motion  of  a  Quadruped.  —  Rope  Dancing.  —  Centre  of  Gravity  of 
two  Bodies  separated  from  each  other.  —  Mathematical  and  experimental 
Examples.  —  The  Conservation  of  the  Motion  of  the  Centre  of  Gravity. — 
Solar  System.  —  Centre  of  Gravity  sometimes  called  Centre  of  Inertia.  .  90 

CHAPTER  X. 

THE  MECHANICAL  PROPERTIES  OF  AN  AXIS. 

An  Axis.  — Planets  and  common  spinning  Top.  — Oscillation  or  Vibration.  — 
Instantaneous  and  continued  Forces. — Percussion.  —  Continued  Force. — 
Rotation.  —  Impressed  Forces.  —  Properties  of  a  fixed  Axis  difficult. — 
Movement  of  the  Force  round  the  Axis.  —  Leverage  of  the  Force.  —  Impulse 
perpendicular  to,  but  not  crossing,  the  Axis.  —  Radius  of  Gyration.  — Cen- 
tre of  Gyration. — Moment  of  Inertia.  —  Principal  Axes.  —  Centre  of  Per- 
cussion  • 108 


CHAPTER  XI. 

OF   THE   PENDULUM. 

Isochronism.  —  Experiments.  —  Simple  Pendulum.  —  Examples  illustrative  of 
—  Length  of.  —  Experiments  of  Kater,  Biot,  Sabine,  and  others.  — Huy- 

gens's  C vcloidal  Pendulum 123 

1* 


*1  CONTENTS. 

CHAPTER  XII. 

OF  SIMPLE  MACHINES. 

Statics.  —  Dynamics.  —  Force.  —  Power.  —  Weight.  —  Lever.  —  Cord.  —  In- 
clined Plane  ....................................................  135 


CHAPTER  XIII. 

OF  THE  LEVER. 

Arms.  —  Fulcrum.  —  Three  Kinds  of  Levers.  —  Crow-Bar.  —  Handspike.  — 
Oar.  —  Nutcrackers.  —  Turning  Lathe.  —  Steelyard  —  Rectangular  Lever. 
Hammer.  —  Load  between  two  Bearers.  —  Combination  of  Levers.  — 
Equivalent  Lever  ..................................  ....  .........  141 

CHAPTER  XIV. 

OF  WHEEL-WORK. 

Wheel  and  Axle.  —  Thickness  of  the  Rope.  —  Ways  of  applying  the  Power. 

—  Projecting  Pins.  —  Windlass.  —  Winch.  —  Axle.  —  Horizontal  Wheel.  — 
Tread-Mill.  —  Cranes.  —  Water-  Wheels.  —  Paddle-  Wheel.  —  Ratchet- 
Wheel.  —  Rack.  —  Spring  of  a  Watch.  —  Fusee.  —  Straps  or  Cords.  — 
Examples  of.  —  Turning  Lathe.  —  Revolving  Shafts.  —  Spinning  Machinery. 

—  Saw-Mill.  —  Pinion.  —  Leaves.  —  Crane.  —  Spur-  Wheels.  —  Crown- 
Wheels.  —  Bevelled   Wheels.  —  Hunting-Cog.  —  Chronometers.  —  Hair- 
Spring.  —  Balance-  Wheel  .......................................  149 

CHAPTER  XV. 

OF  THE  PULLEY. 

Cord:  —  Sheave.  —  Fixed  Pulley.  —  Fire  Escapes.  —  Single  movable  Pul- 
ley. —  Systems  of  Pulleys.  —  Smeaton's  Tackle.  —  White's  Pulley.  —  Ad- 
vantage of.  —  Runner.  —  Spanish  Bartons  ........................  166 

CHAPTER  XVI. 

ON  THE  INCLINED  PLANE,  WEDGE,  AND  SCREW. 

Inclined  Plane.  —  Effect  of  a  Weight  on.  —  Power  of.  —  Roads.  —  Power 
oblique  to  the  Plane.  —  Plane  sometimes  moves  under  the  Weight.  — 
Wedge.  —  Sometimes  formed  of  two  inclined  Planes.  —  More  powerful  as 
its  Angle  is  acute.  —  Where  used.  —  Limits  to  the  Angle.  —  Screw.  — 
Hunter's  Screw.  —  Examples.  —  Micrometer  Screw  ................  176 


CONTENTS.  Yll 

CHAPTER  XVII. 

ON  THE  REGULATION  AND  ACCUMULATION  OF  FORCE. 

Uniformity  of  Operation.  —  Irregularity  of  prime  Mover.  —  Water-Mill.  — 
Wind-Mill.  —  Steam  Pressure.  —  Animal  Power.  —  Spring.  —  Regulators. 

—  Steam-Engine.  —  Governor.  —  Self-acting  Damper.  —  Tachometer.  — 
Accumulation  of  Power.  —  Examples.  —  Hammer.  —  Flail.  —  Bow-String. 

—  Fire- Arms.  —  Air-Gun.  — Steam-Gun. —  Inert  Matter  a  Magazine  for 
Force.  —  Fly- Wheel.  —  Condensed  Air.  —  Rolling  Metal.  —  Coining- 
Press 189 


CHAPTER  XVIII. 


MECHANICAL  CONTRIVANCES  FOR  MODIFYING  MOTION. 

Division  of  Motion  into  rectilinear  and  rotatory.  —  Continued  and  reciprocat- 
ing. —  Examples.  —  Flowing  Water.  —  Wind.  —  Animal  Motion.  —  Fall- 
ing of  a  Body.  —  Syringe-Pump.  —  Hammer.  —  Steam-Engine.  —  Fulling- 
Mill.  —  Rose-Engine.  —  Apparatus  of  Zureda.  —  Leupold's  Application 
of  it.  —  Hooke's  universal  Joint.  —  Circular  and  alternate  Motion.  —  Ex- 
amples. —  Watt's  Methods  of  connecting  the  Motion  of  the  Piston  with  that 
of  the  Beam.  —  Parallel  Motion 20t, 


CHAPTER  XIX. 


OF  FRICTION  AND  THE  RIGIDITY  OF  CORDAGE. 

Friction  and  Rigidity.  —  Laws  of  Friction.  —  Rigidity  of  Cordage.  —  Strength 
of  Materials.  —  Resistance  from  Friction.  —  Independent  of  the  Magnitude 
of  Surfaces.  —  Examples.  —  Vince's  Experiments. —  Effect  of  Velocity. — 
Means  for  diminishing  Friction.  —  Friction-Wheels.  —  Angle  of  Repose.  — 
Best  Angle  of  Draught.  —  Rail-Roads.  —  Stiffness  of  Ropes 219 


CHAPTER  XX. 

ON  THE  STRENGTH  OF  MATERIALS. 

Difficulty  of  determining  the  Laws  which  govern  the  Strength  of  Materials.  — 
Forces  tending  to  separate  the  Parts  of  a  Solid.  —  Laws  by  which  Solids 
resist  Compression.  —  Euler's  Theory.  —  Transverse  Strength  of  Solids.  — 
Strength  diminished  by  the  Increase  of  Height.  —  Lateral  or  Transverse 
Strain.  —  Limits  of  Magnitude.  —  Relative  Strength  of  small  Animals 
greater  than  large  ones 229 


Vlli  CONTENTS. 

CHAPTER  XXI. 

ON  BALANCES  AND  PENDULUMS. 

Weight,  —  Time.  —  The  Balance.  —  Fulcrum.  —  Centre  of  Gravity  of.  — 
Sensibility  of.  —  Positions  of  the  Fulcrum.  —  Beam  variously  constructed. 
—  Troughton's  Balance.  —  Robinson's  Balance.  —  Rater's  Balance.  — 
Method  of  adjusting  a  Balance.  —  Use  of  it.  —  Precautions  necessary.  — 
Of  Weights.  —  Adjustment  of.  —  Dr.  Black's  Balance.  —  Steelyard.  — 
Roman  Statera  or  Steelyard.  —  Convenience  of.  —  C.  Paul's  Steelyard.  — 
Chinese  Steelyard.  —  Danish  Balance.  —  Bent  Lever  Balance.  —  Brady's 
Balance.  —  Weighing  Mackine  for  Turnpike  Roads.  —  Instruments  for 
Weighing  by  Means  of  a  Spring.  —  Spring  Steelyard.  —  Sailer's  Spring 
Balance.  —  Marriott's  Dial  Weighing  Machine.  —  Dynamometer.  —  Com- 
pensation Pendulums.  —  Barton's  Gridiron  Pendulum.  —  Table  of  linear 
Expansion.  —  Second  Table.  —  Harrison's  Pendulum.  —  Troughton's  Pen- 
dulum. —  Benzenberg's  Pendulum.  —  Ward's  Compensation  Pendulum.  — 
Compensation  Tube  of  Julien  le  Roy.  —  Deparcieux's  Compensation.  — 
Kater's  Pendulum.  — Reed's  Pendulum.  —  Ellicott's  Pendulum.  — Mercurial 
Pendulum.  —  Graham's  Pendulum.  —  Compensation  Pendulum  of  Wood 
and  Lead.  —  Smeaton's  Pendulum.  —  Brown's  Mode  of  Adjustment. ..  234 


THE 


ELEMENTS   OF  MECHANICS. 


CHAPTER  I, 

PROPERTIES    OP    MATTER MAGNITUDE IMPENETRABILITY 

FIGURE FORCE. 

(1.)  PLACED  in  the  material  world,  Man  is  continually 
exposed  to  the  action  of  an  infinite  variety  of  objects  by 
which  he  is  surrounded.  The  body,  to  which  the  thinking 
and  living  principles  have  been  united,  is  an  apparatus  ex- 
quisitely contrived  to  receive  and  to  transmit  these  impres- 
sions. Its  various  parts  are  organized  with  obvious  reference 
to  the  several  external  agents  by  which  it  is  to  be  affected. 
Each  organ  is  designed  to  convey  to  the  mind  immediate 
notice  of  some  peculiar  action,  and  is  accordingly  endued 
with  a  corresponding  susceptibility-.  This  adaptation  of  the 
organs  of  sense  to  the  particular  influences  of  material  agents, 
is  rendered  still  more  conspicuous  when  we  consider  that, 
however  delicate  its  structure,  each  organ  is  wholly  insensible 
to  every  influence  except  that  to  which  it  appears  to  be 
specially  appropriated.  The  eye,  so  intensely  susceptible  of 
impressions  from  light,  is  not  at  all  affected  by  those  of  sound ; 
while  the  fine  mechanism  of  the  ear,  so  sensitively  alive  to 
every  effect  of  the  latter  class,  is  altogether  insensible  to  the 
former.  The  splendor  of  excessive  light  may  occasion  blind- 
ness, and  deafness  may  result  from  the  roar  of  a  cannonade ; 
but  neither  the  sight  nor  the  hearing  can  be  injured  by  the 
most  extreme  action  of  that  principle  which  is  designed  to 
affect  the  other. 

Thus  the  organs  of  sense  are  instruments  by  which  the 
mind  is  enabled  to  determine  the  existence  and  the  qualities 
of  external  things.  The  effects  which  these  objects  produce 
upon  the  mind  through  the  organs,  are  called  sensations,  and 
these  sensations  are  the  immediate  elements  of  all  human 


3  THE    ELEMENTS    OF    MECHANICS.  CHAP.  I 

knowledge.  MATTER  is  the  general  name  which  has  been 
given  to  that  substance,  which,  under  forms  infinitely  various, 
affects  the  senses.  Metaphysicians  have  differed  in  defining 
this  principle.  Some  have  even  doubted  of  its  existence. 
But  these  discussions  are  beyond  the  sphere  of  mechanical 
philosophy,  the  conclusions  of  which  are  in  no  wise  affected 
by  them.  Our  investigations  here  relate,  not  to  matter  as  an 
abstract  existence,  but  to  those  qualities  which  we  discover  in 
it  by  the  senses,  and  of  the  existence  of  which  we  are  sure, 
however  the  question  as  to  matter  itself  may  be  decided. 
When  we  speak  of"  bodies,"  we  mean  those  things,  whatever 
they  be,  which  excite  in  our  minds  certain  sensations  ;  and 
the  powers  to  excite  those  sensations  are  called  "  properties," 
or  "qualities." 

(2.)  To  ascertain,  by  observation,  the  properties  of  bodies, 
is  the  first  step  towards  obtaining  a  knowledge  of  nature. 
Hence  man  becomes  a  natural  philosopher  the  moment  he 
begins  to  feel  and  to  perceive.  The  first  stage  of  life  is  a 
state  of  constant  and  curious  excitement.  Observation  and 
attention,  ever  awake,  are  engaged  upon  a  succession  of 
objects  new  and  wonderful.  The  large  repository  of  the 
memory  is  opened,  and  every  hour  pours  into  it  unbounded 
stores  of  natural  facts  and  appearances,  the  rich  materials  of 
future  knowledge.  The  keen  appetite  for  discovery,  implant- 
ed in  the  mind  for  the  highest  ends,  continually  stimulated 
by  the  presence  of  what  is  novel,  renders  torpid  every  other 
faculty,  and  the  powers  of  reflection  and  comparison  are  lost 
in  the  incessant  activity  and  unexhausted  vigor  of  observa- 
tion. After  a  season,  however,  the  more  ordinary  classes 
of  phenomena  cease  to  excite  by  their  novelty.  Attention 
is  drawn  from  the  discovery  of  what  is  new,  to  the  examina- 
tion of  what  is  familiar.  From  the  external  world  the  mind 
turns  in  upon  itself,  and  the  feverish  astonishment  of  child- 
hood gives  place  to  the  more  calm  contemplation  of  incipient 
maturity.  The  vast  and  heterogeneous  mass  of  phenomena 
collected  by  past  experience  is  brought  under  review.  The 
great  work  of  comparison  begins.  Memory  produces  her 
stores,  and  reason  arranges  them.  Then  succeed  those  first 
attempts  at  generalization  which  mark  the  dawn  of  science 
in  the  mind. 

To  compare,  to  classify,  to  generalize,  seem  to  be  instinc- 
tive propensities  peculiar  to  man.  They  separate  him  from 
inferior  animals  by  a  wide  chasm.  It  is  to  these  powers  that 


CHAP.  I.  MAGNITUDE.  3 

all  the  higher  mental  attributes  may  be  traced,  and  it  is  from 
their  right  application  that  all  progress  in  science  must  arise. 
Without  these  powers,  the  phenomena  of  nature  would 
continue  a  confused  heap  of  crude  facts,  with  which 
the  memory  might  be  loaded,  but  from  which  the  intellect 
would  derive  no  advantage.  Comparison  and  generalization 
are  the  great  digestive  organs  of  the  mind,  by  which  only 
nutrition  can  be  extracted  from  this  mass  of  intellectual  food, 
and  without  which  observation  the  rrtfest  extensive,  and  atten- 
tion the  most  unremitting,  can  be  productive  of  no  real  or 
useful  advancement  in  knowledge. 

(3.)  Upon  reviewing  those  properties  of  bodies  which  the 
senses  most  frequently  present  to  us,  we  observe  that  very 
few  of  them  are  essential  to,  and  inseparable  from,  matter. 
The  greater  number  may  be  called  particular  or  peculiar 
qualities,  being  found  in  some  bodies,  but  not  in  others. 
Thus  the  property  of  attracting  iron  is  peculiar  to  the  load- 
stone, and  not  observable  in  other  substances.  One  body 
excites  the  sensation  of  green,  another  of  red,  and  a  third 
is  deprived  of  all  color.  A  few  characteristic  and  essential 
qualities  are,  however,  inseparable  from  matter  in  whatever 
state,  or  under  whatever  form  it  exist.  Such  properties  alone 
can  be  considered  as  tests  of  materiality.  Where  their  pres- 
ence is  neither  manifest  to  sense,  nor  demonstrable  by  reason, 
there  matter  is  not.  The  principal  of  these  qualities  are 
magnitude  and  impenetrability. 

(4.)  Magnitude. — Every  body  occupies  space ;  that  is,  it 
has  magnitude.  This  is  a  property  observable  by  the  senses  in 
all  bodies  which  are  not  so  minute  as  to  elude  them,  and  which 
the  understanding  can  trace  to  the  smallest  particle  of  matter. 
It  is  impossible,  by  any  stretch  of  imagination,  even  to  con- 
ceive a  portion  of  matter  so  minute  as  to  have  no  magnitude. 

The  quantity  of  space  which  a  body  occupies  is  sometimes 
called  its  magnitude.  In  colloquial  phraseology,  the  word 
size  is  used  to  express  this  notion ;  but  the  most  correct  term, 
and  that  which  we  shall  generally  adopt,  is  volume.  Thus 
we  say,  the  volume  of  the  earth  is  so  many  cubic  miles,  the 
volume  of  this  room  is  so  many  cubic  feet. 

The  external  limits  of  the  magnitude  of  a  body  are  lines 
and  surfaces,  lines  being  the  limits  which  separate  the  several 
surfaces  of  the  same  body.  The  linear  limits  of  a  body  are 
also  called  edges.  Thus  the  line  which  separates  the  top  of  a 
chest  from  one  of  its  sides  is  called  an  edge. 


4  THE  ELEMENTS  OP  MECHANICS.  CHAP.  I 

The  quantity  of  a  surface  is  called  its  area,  and  the  quan- 
tity of  a  line  is  called  its  length.  Thus  we  say,  the  area  of  a 
field  is  so  many  acres,  the  length  of  a  rope  is  so  many  yards. 
The  word  "  magnitude"  is,  however,  often  used  indifferently 
for  volume,  area,  and  length.  If  the  objects  of  investigation 
were  of  a  more  complex  and  subtle  character,  as  in  metaphys- 
ics, this  unsteady  application  of  terms  might  be  productive 
of  confusion,  and  even  of  error  ;  but  in  this  science,  the  mean- 
ing of  the  term  is  evident,  from  the  way  in  which  it  is  ap- 
plied, and  no  inconvenience  is  found  to  arise. 

(5.)  Impenetrability. — This  property  will  be  most  clearly 
explained  by  defining  the  positive  quality  from  which  it  lakes 
its  name,  and  of  which  it  merely  signifies  the  absence.  A 
substance  would  be  penetrable  if  it  were  such  as  to  allow 
another  to  pass  through  the  space  which  it  occupies,  without 
disturbing  its  component  parts.  Thus,  if  a  comet,  striking 
the  earth,  could  enter  it  at  one  side,  and,  passing  through  it, 
emerge  from  the  other  without  separating  or  deranging  any 
bodies  on  or  within  the  earth,  then  the  earth  would  be  pene- 
trable by  the  comet.  When  bodies  are  said  to  be  impenetra- 
ble, it  is  therefore  meant  that  one  cannot  pass  through  another 
without  displacing  some  or  all  of  the  component  parts  of  that 
other.  There  are  many  instances  of  apparent  penetration  ; 
but  in  all  these,  the  parts  of  the  body  which  seem  to  be  pene- 
trated are  displaced.  Thus,  if  tV.e  point  of  a  needle  be  plung- 
ed in  a  vessel  of  water,  all  the  water  which  previously  filled 
the  space  into  which  the  needle  enters  will  be  displaced,  and 
the  level  of  the  water  will  rise  in  the  vessel  to  the  same  height 
as  it  would  by  pouring  in  so  much  more  water  as  would  fill 
the  space  occupied  by  the  needle. 

(6.)  Figure. — If  the  hand  be  placed  upon  a  solid  body, 
we  become  sensible  of  its  impenetrability,  by  the  obstruction 
which  it  opposes  to  the  entrance  of  the  hand  within  its  di- 
mensions. We  are  also  sensible  that  this  obstruction  com- 
mences at  certain  places;  that  it  has  certain  determinate 
limits  ;  that  these  limitations  are  placed  in  certain  directions 
relatively  to  each  other.  The  mutual  relation  which  is  found 
to  subsist  between  these  boundaries  of  a  body,  gives  us  the 
notion  of  its  figure.  The  figure  and  volume  of  a  body  should 
be  carefully  distinguished.  Each  is  entirely  independent 
of  the  other.  Bodies  having  very  different  volumes  may  have 
the  same  figure  ;  and  in  like  manner  bodies  differing  in  fig- 
ure may  have  the  same  volume.  The  figure  of  a  body  is 


CHAP.  I.  FORCE.  5 

what,  in  popular  language,  is  called  its  shape,  or  form.  The 
volume  of  a  body  is  that  which  is  commonly  called  its  size. 
It  will  hence  be  easily  understood,  that  one  body  (a  globe, 
for  example)  may  have  ten  times  the  volume  of  another  (globe), 
and  yet  have  the  same  figure ;  and  that  two  bodies  (as  a  die 
and  a  globe)  may  have  jigvrts  altogether  different,  and  yet  have 
equal  volume*.  What  we  have  here  observed  of  volumes  will 
also  be  applicable  to  lengths  and  areas.  The  arc  of  a  circle 
and  a  straight  line  may  have  the  same  length,  although  they 
have  different  figures ;  and,  on  the  other  hand,  two  arcs  of 
different  circles  may  have  the  same  figure,  but  very  unequal 
lengths.  The  surface  of  a  ball  is  curved,  that  of  the  table 
plane  ;  and  yet  the  area  of  the  surface  of  the  ball  may  be 
equal  to  that  of  the  table. 

(7.)  Atoms — Molecule*. — Impenetrability  must  not  be  con- 
founded with  inseparability.  Every  body  which  has  been 
brought  under  human  observation  is  separable  into  parts  , 
and  these  parts,  however  small,  are  separable  into  others  still 
more  minute.  To  this  process  of  division  no  practical  limit 
has  ever  been  found.  Nevertheless,  many  of  the  phenomena 
which  the  researches  of  those  who  have  successfully  examined 
the  laws  of  nature  have  developed,  render  it  highly  probable 
that  all  bodies  are  composed  of  elementary  parts  which  are 
indivisible  and  unalterable.  The  component  parts,  which 
may  be  called  atoms,  are  so  minute  as  altogether  to  elude 
the  senses,  even  when  improved  by  the  most  powerful  aids 
of  art.  The  word  molecule  is  often  used  to  signify  component 
parts  of  a  body  so  small  as  to  escape  sensible  observation, 
but  not  ultimate  atoms,  each  molecule  being  supposed  to  be 
{ormed  of  several  atoms,  arranged  according  to  some  deter- 
minate figure.  Particle  is  used  also  to  express  small  compo- 
nent parts,  but  more  generally  is  applied  to  those  which  are 
not  too  minute  to  be  discoverable  by  observation. 

(8.)  force. — If  the  particles  of  matter  were  endued  with 
no  property  in  relation  to  one  another,  except  their  mutual 
impenetrability,  the  universe  would  be  like  a  mass  of  sand, 
without  variety  of  state  or  form.  Atoms,  when  placed  in 
juxtaposition,  would  neither  cohere,  as  in  solid  bodies,  nor  repel 
each  other,  as  in  aeriform  substances.  We  find,  on  the  other 
hand,  that,  in  some  cases,  the  atoms  which  compose  bodies 
are  not  simply  placed  together,  but  a  certain  effect  is  mani- 
fested in  their  strong  coherence.  If  they  were  merely  placed 
in  juxtaposition,  their  separation  would  be  effected  as  easily 
1  * 


O  THE    ELEMENTS    OF    MECHANICS.  CHAP.    I. 

3jj*    '«-. . 

as  any  component  particle  could  be  removed  from  one 
place  to  another.  Take  a  piece  of  iron,  and  attempt  to  sep- 
arate its  parts  :  the  effort  will  be  strongly  resisted,  and  it  will 
be  a  matter  of  much  greater  facility  to  remove  the  whole 
mass.  It  appears,  therefore,  that,  in  such  cases,  the  parts 
which  are  in  juxtaposition  cohere,  and  resist  their  mutual 
separation.  This  effect  is  denominated  force  ;  and  tfce  con- 
stituent atoms  are  said  to  cohere  with  a  greater  or  less  degree 
of  force,  according  as  they  oppose  a  greater  or  less  resistance 
to  their  mutual  separation. 

The  coherence  of  particles  in  juxtaposition  is  an  effect 
of  the  same  class  as  the  mutual  approach  of  particles  placed 
at  a  distance  from  each  other.  It  is  not  difficult  to  perceive 
that  the  same  influence  which  causes  the  bodies  A  and  B  to 
approach  each  other,  when  placed  at  some  distance  asunder, 
will,  when  they  unite,  retain  them  together,  and  oppose  a  resist- 
ance to  their  separation.  Ilejice  this  effect  of  the  mutual  ap- 
proximation of  bodies  towards  each  other  is  also  called  fore  e. 

Force  is  generally  denned  to  be  "  whatever  produces  or 
opposes  the  production  of  motion  in  matter."  In  this  sense, 
it  is  a  name  for  the  unknown  cause  of  a  known  effect.  It 
would,  however,  be  more  philosophical  to  give  the  name,  not 
to  the  cause,  of  which  we  are  ignorant,  but  to  the  effect,  of 
which  we  have  sensible  evidence.  To  observe  and  to  classify 
is  the  whole  business  of  the  natural  philosopher.  When 
causes  are  referred  to,  it  is  implied,  that  effects  of  the  same 
class  arise  from  the  agency  of  the  same  cause.  However 
probable  this  assumption  may  be,  it  is  altogether  unnecessary. 
All  the  objects  of  science,  the  enlargement  of  mind,  the  ex- 
tension and  improvement  of  knowledge,  the  facility  of  its 
acquisition,  are  obtained  by  generalization  alone,  and  no 
good  can  arise  from  tainting  our  conclusions  with  the  possible 
errors  of  hypotheses. 

It  may  be  here,  once  for  all,  observed,  that  the  phraseology 
of  causation  and  hypotheses  has  become  so  interwoven  with 
the  language  of  science,  that  it  is  impossible  to  avoid  the  fre- 
quent use  of  it.  Thus  we  say,  "  the  magnet  attracts  iron  ;" 
the  expression  attract  intimating  the  cause  of  the  observed 
effect.  In  such  cases,  however,  we  must  be  understood 
to  mean  the  effect  itself,  finding  it  less  inconvenient  to  con- 
tinue the  use  of  the  received  phrases,  modifying  their  signifi- 
cation, than  to  introduce  new  ones. 

Force,  when  manifested  by  the  mutual  approach  or  cohe- 


CHAP.    II.  PROPERTIES    OF    MATTER.  7 

sion  of  bodies,  is  also  called  attraction,  and  it  is  variously 
denominated,  according  to  the  circumstances  under  which  it 
is  observed  to  act.  Thus  the  force  which  holds  together  the 
atoms  of  solid  bodied  is  called  cohesive  attraction.  The  force 
which  draws  bodies  to  the  surface  of  the  earth,  when  placed 
above  it,  is  called  the  attraction  of  gravitation.  The  force 
which  is  exhibited  by  the  mutual  approach,  or  adhesion  of  the 
loadstone  and  iron,  is  called  magnetic  attraction;  and  so  on. 

When  force  is  manifested  by  the  remotion  of  bodies  from 
each  other,  it  is  called  repnhion.  Thus,  if  a  piece  of  glass, 
having  been  briskly  rubbed  with  a  silk  handkerchief,  touch, 
successively,  two  feathers,  these  feathers,  if  brought  near 
each  other,  will  move  asunder.  This  effect  is  called  repul- 
sion, and  the  feathers  are  said  to  repel  each  other. 

(9.)  The  influence  which  forces  have  upon  the  form,  state, 
arrangement  and  motions  of  material  substances,  is  the 
principal  object  of  physical  science.  In  its  strict  sense,  ME- 
CHANICS is  a  term  of  very  extensive  signification.  According 
to  the  more  popular  usage,  however,  it  has  been  generally 
applied  to  that  part  of  physical  science  which  includes  the 
investigation  of  the  phenomeni  of  motion  and  rest,  pressure, 
and  other  effects  developed  by  the  mutual  action  of  solid 
masses.  The  consideration  of  similar  phenomena,  exhibited 
in  bodies  of  the  liquid  form,  is  consigned  to  HYDROSTATICS, 
and  that  of  aeriform  fluids  to  PNEUMATICS. 


CHAPTER   II. 

DIVISIBILITY POROSITY DENSITY COMPRESSIBILITY ELAS- 
TICITY  DILATABILITY. 

(10.)  BESIDES  the  qualities,  magnitude  and  impenetra- 
bility, there  .are  several  other  general  properties  of  bodies 
contemplated  in  mechanical  philosophy,  and  to  which  we 
shall  have  frequent  occasion  to  refer.  Those  which  we  shall 
notice  in  the  present  chapter  are, 

1.  Divisibility. 

2.  Porosity — Density. 

3.  Compressibility — Elasticity. 

4.  Dilatability. 

(11.)  Divisibility. — Observation  and  experience  prove  that 


8  THE  ELEMENTS  OF  MECHANICS.      CHAP.  II. 

all  bodies  of  sensible  magnitude,  even  the  most  solid,  consist 
of  parts  which  are  separable.  To  the  practical  subdivision 
of  matter  there  seems  to  be  no  assignable  limit.  Numerous 
examples  of  the  division  of  matter,  to  a  degree  almost  ex- 
ceeding belief,  may  be  found  in  experimental  inquiries  insti- 
tuted in  physical  science  ;  the  useful  arts  furnish  many  in- 
stances not  less  striking ;  but,  perhaps,  the  most  conspicuous 
proofs  which  can  be  produced,  of  the  extreme  minuteness 
of  which  the  parts  of  matter  are  susceptible,  arise  from  the 
consideration  of  certain  parts  of  the  organized  world. 

(12.)  The  relative  places  of  stars  in  the  heavens,  as  seen 
in  the  field  of  view  of  a  telescope,  are  marked  by  fine  lines 
of  wire  placed  before  the  eye-glass,  and  which  cross  each 
other  at  right  angles.  The  stars  appearing  in  the  telescope 
as  mere  lucid  points  without  sensible  magnitude,  it  is  neces- 
sary that  the  wires  which  mark  their  places  should  have  a 
corresponding  tenuity.  But  these  wires,  being  magnified  by 
the  eye-glass,  would  have  an  apparent  thickness,  which 
would  render  them  inapplicable  to  this  purpose,  unless  their 
real  dimensions  were  of  a  most  uncommon  degree  of  minute- 
ness. To  obtain  wire  for  this  purpose,  Dr.  Wollaston  invent- 
ed the  following  process  : — A  piece  of  fine  platinum  wire,  a 
b,  is  extended  along  the  axis  of  a  cylindrical  mould.  A  B,^. 
1.  Into  this  mould,  at  A,  molten  silver  is  poured.  Since 
the  heat  necessary  for  the  fusion  of  platinum  is  much  greater 
than  that  which  retains  silver  in  the  liquid  form,  the  wire  a  b 
remains  solid,  while  the  mould  A  B  is  filled  with  the  silver. 
When  the  metal  has  become  solid  by  being  cooled,  and  has 
been  removed  from  the  mould,  a  cylindrical  bar  of  silver  is 
obtained,  having  a  platinum  wire  in  its  axis.  This  bar  is 
then  wire-drawn,  by  forcing  it  successively  through  holes  C, 
D,  E,  F,  G,  H,  diminishing  in  magnitude,  the  first  hole 
being  a  little  less  than  the  wire  at  the  beginning  of  the 
process.  By  these  means  the  platinum  a  b  is  wire-drawn  at 
the  same  time,  and  in  the  same  proportion  with*  the  silver,  so 
that,  whatever  be  the  original  proportion  of  the  thickness  of 
the  wire  a  b  to  that  of  the  mould  A  B,  the  same  will  be  the 
proportion  of*  the  platinum  wire  to  the  whole  at  the  several 
thicknesses  C,  D,  &/c.  If  we  suppose  the  mould  A  B  to  be 
ten  times  the  thickness  of  the  wire  a  b,  then  the  silver  wire, 
throughout  the  whole  process,  will  be  ten  times  the  thickness 
of  the  platinum  wire  which  it  includes  within  it.  The  silver 
wire  may  be  drawn  to  a  thickness  not  exceeding  the  300th  of 


CHAP.  II.  DIVISIBILITY.  9 

an  inch.  The  platinum  will  thus  not  exceed  the  3000th  of  an 
inch.  The  wire  is  then  dipped  in  nitric  acid,  which  dissolves 
the  silver,  but  leaves  the  platinum  solid.  By  this  method 
Dr.  Wollaston  succeeded  in  obtaining  wire,  the  diameter 
of  which  did  not  exceed  the  18,000th  of  an  inch.  A  quantity 
of  this  wire,  equal  in  bulk  to  a  common  die  used  in  games 
of  chance,  would  extend  from  Paris  to  Rome. 

(13.)  Newton  succeeded  in  determining  the  thickness  of 
very  thin  laminae  of  transparent  substances  by  observing  the 
colors  which  they  reflect.  A  soap  bubble  is  a  thin  shell 
of  water,  and  is  observed  to  reflect  different  colors  from  dif- 
ferent parts  of  its  surface.  Immediately  before  the  bubble 
bursts,  a  black  spot  may  be  observed  near  the  top.  At 
this  part  the  thickness  has  been  proved  not  to  exceed  the 
2,500,000th  of  an  inch. 

The  transparent  wings  of  certain  insects  are  so  attenuated 
in  their  structure,  that  50,000  of  them  placed  over  each  other 
would  not  form  a  pile  a  quarter  of  an  inch  in  height. 

(14.)  In  the  manufacture  of  embroidery,  it  is  necessary  to 
obtain  very  fine  gilt  silver  threads.  To  accomplish  this,  a 
cylindrical  bar  of  silver,  weighing  360  ounces,  is  covered 
with  about  two  ounces  of  gold.  This  gilt  bar  is  then  wire- 
drawn, as  in  the  first  example,  until  it  is  reduced  to  a  thread 
so  fine  that  3400  feet  of  it  weigh  less  than  an  ounce.  The 
wire  is  then  flattened  by  passing  it  between  rollers  under 
a  severe  pressure — a  process  which  increases  its  length,  so 
that  about  4000  feet  shall  weigh  one  ounce.  Hence  one  foot 
will  weigh  the  4000th  part  of  an  ounce.  The  proportion  of 
the  gold  to  the  silver  in  the  original  bar  was  that  of  2  to  360, 
or  1  to  180.  Since  the  same  proportion  is  preserved  after  the 
bar  has  been  wire-drawn,  it  follows  that  the  quantity  of  gold 
which  covers  one  foot  of  the  fine  wire  is  the  180th  part  of  the 
4000th  of  an  ounce  ;  that  is,  the  720,000th  part  of  an  ounce. 

The  quantity  of  gold  which  covers  one  inch  of  this  wire 
will  be  twelve  times  less  than  that  which  covers  one  foot. 
Hence  this  quantity  will  be  the  8,640,000th  part  of  an  ounce. 
If  this  inch  be  again  divided  into  100  equal  parts,  every  part 
will  be  distinctly  visible  without  the  aid  of  microscopes. 
The  gold  which  covers  this  small  but  visible  portion  is  the 
864,000,000th  part  of  an  ounce.  But  we  may  proceed  even 
further ;  this  portion  of  the  wire  may  be  viewed  by  a  micro- 
scope which  magnifies  500  times,  so  that  the  500th  part 
of  it  will  thus  become  visible.  In  this  manner,  therefore, 


10  THE  ELEMENTS  OF  MECHANICS.       CHAP.  II. 

an  ounce  of  gold  may  be  divided  into  432,000,000,000  parts. 
Each  of  these  parts  will  possess  all  the  characters  and  quali- 
ties which  are  found  in  the  largest  masses  of  the  metal.  It 
retains  its  solidity,  texture,  and  color ;  it  resists  the  same 
agents,  and  enters  into  combination  with  the  same  substances. 
If  the  gilt  wire  be  dipped  in  nitric  acid,  the  silver  within  the 
coating  will  be  dissolved,  but  the  hollow  tube  of  gold  which 
surrounded  it  will  still  cohere  and  remain  suspended. 

(15.)  The  organized  world  offers  still  more  remarkable 
examples  of  the  inconceivable  subtilty  of  matter. 

The  blood  which  flows  in  the  veins  of  animals  is  not,  as  it 
seems,  an  uniformly  red  liquid.  It  consists  of  small  red 
globules,  floating  in  a  transparent  fluid  called  scrum.  In 
different  species  these  globules  differ  both  in  figure  and 
in  magnitude.  ^In  man  and  all  animals  which  suckle  their 
young,  they  are  perfectly  round  or  spherical.  In  birds  an4 
fishes,  they  are  of  an  oblong  spheroidal  form.  In  the  human 
species,  the  diameter  of  the  globules  is  about  the  4000th 
of  an  inch.  Hence  it  follows,  that  in  a  drop  of  blood  which 
would  remain  suspended  from  the  point  of  a  fine  needle, 
there  must  be  about  a  million  of  globules. 

Small  as  these  globules  are,  the  animal  kingdom  presents 
beings  whose  whole  bodies  are  still  more  minute.  Animal- 
cules have  been  discovered,  whose  magnitude  is  such,  that 
a  million  of  them  does  not  exceed  the  bulk  of  a  grain  of 
sand  ;  and  yet  each  of  these  creatures  is  composed  of  mem- 
bers as  curiously  organized  as  those  of  the  largest  species ; 
they  have  <life  and  spontaneous  motion,  and  are  endued  with 
sense  and  instinct.  In  the  liquids  in  which  they  live,  they 
are  observed  to  move  with  astonishing  speed  and  activity ;  nor 
are  their  motions  blind  and  fortuitous,  but  evidently  governed 
by  choice,  and  directed  to  an  end.  They  use  food  and  drink, 
from  which  they  derive  nutrition,  and  are  therefore  furnished 
with  a  digestive  apparatus.  They  have  great  muscular  power, 
and  are  furnished  with  limbs  and  muscles  of  strength  and  flexi- 
bility. They  are  susceptible  of  the  same  appetites,  and  obnox- 
ious to  the  same  passions,  the  gratification  of  which  is  attend- 
ed with  the  same  results  as  in  our  own  species.  Spallanzani 
observes,  that  certain  animalcules  devour  others  so  voraciously, 
that  they  fatten  and  become  indolent  and  sluggish  by  over- 
feeding. After  a  meal  of  this  kind,  if  they  be  confined  in 
distilled  water,  so  as  to  be  deprived  of  all  food,  their  condition 
becomes  reduced  ;  they  regain  their  spirit  and  activity,  and 


CHAP.  II.  ULTIMATE  ATOMS.  11 

amuse  themselves  in  the  pursuit  of  the  more  minute  animals, 
which  are  supplied  to  them  ;  they  swallow  these  without 
depriving  them  of  life,  for,  by  the  aid  of  the  microscope, 
the  one  has  been  observed  moving  within  the  body  of  the 
other.  These  singular  appearances  are  not  matters  of  idle 
and  curious  observation.  They  lead  us  to  inquire  what 
parts  are  necessary  to  produce  such  results.  Must  we  not 
conclude  that  these  creatures  have  heart,  arteries,  veins,  mus- 
cles, sinews,  tendons,  nerves,  circulating  fluids,  and  all  the 
concomitant  apparatus  of  a  living  organized  body  ?  And 
if  so,  how  inconceivably  minute  must  those  parts  be  !  If  a 
globule  of  their  blood  bears  the  same  proportion  to  their 
whole  bulk  as  a  globule  of  our  blood  bears  to  our  magnitude, 
what  powers  of  calculation  can  give  an  adequate  notion  of  its 
minuteness  ? 

(16.)  These  and  many  other  phenomena  observed  in  the 
immediate  productions  of  nature,  or  developed  by  mechanical 
and  chemical  processes,  prove  that  the  materials  of  which 
bodies  are  formed  are  susceptible  of  minuteness  which  in- 
finitely exceeds  the  powers  of  sensible  observation,  even  when 
those  powers  have  been  extended  by  all  the  aids  of  science. 
Shall  we  then  conclude  that  matter  is  infinitely  divisible, 
and  that  there  are  no  original  constituent  atoms  of  determi- 
nate magnitude  and  figure  at  which  all  subdivision  must 
cease?  Such  an  inference  would  be  unwarranted,  even  had 
we  no  other  means  of  judging  the  question,  except  those 
of  direct  observation  ;  for  it  would  be  imposing  that  limit 
on  the  works  of  nature  which  she  has  placed  upon  our  powers 
of  observing  them.  Aided  by  reason,  however,  and  a  due 
consideration  of  certain  phenomena  which  come  within  our 
immediate  powers  of  observation,  we  are  frequently  able  to 
determine  other  phenomena  which  are  beyond  those  powers. 
The  diurnal  motion  of  the  earth  is  not  perceived  by  us,  be 
cause  all  things  around  us  participate  in  it,  preserve  their 
relative  position,  and  appear  to  be  c.t  rest.  But  reason  tells 
us  that  such  a  motion  must  produce  the  alternations  of  day 
and  night,  and  the  rising  and  setting  of  all  the  heavenly 
bodies — appearances  which  are  plainly  observable,  and  which 
betray  the  cause  from  which  they  arise.  Again,  we  cannot 
place  ourselves  at  a  distance  from  the  earth,  and  behold 
the  axis  on  which  it  revolves,  and  observe  its  peculiar  obli- 
quity to  the  orbit  in  which  the  earth  moves  ;  but  we  see 
and  feel  the  vicissitudes  of  the  seasons — an  effect  which  is  the 


12  THE  ELEMENTS  OF  MECHANICS.       CHAP.  II. 

immedi  te  consequence  of  that  inclination,  and  by  which  we 
are  able  to  detect  it. 

(17.)  So  it  is  in  the  present  case.  Although  we  are 
unable  by  direct  observation  to  prove  the  existence  of  con- 
stituent material  atoms  of  determinate  figure,  yet  there  are 
many  observable  phenomena  which  render  their  existence 
in  the  highest  degree  probable,  if  not  morally  certain.  The 
most  remarkable  of  this  class  of  effects  is  observed  in  the 
crystallization  of  salts.  When  salt  is  dissolved  in  a  sufficient 
quantity  of  pure  water,  it  mixes  with  the  water  in  such  a 
manner  as  wholly  to  disappear  to  the  sight  and  touch, 
the  mixture  being  one  uniform  transparent  liquid,  like  the 
water  itself,  before  its  union  with  the  salt.  The  presence 
of  the  salt  in  the  water  may,  however,  be  ascertained  by 
weighing  the  mixture,  which  will  be  found  to  exceed  the 
original  weight  of  the  water  by  the  exact  amount  of  the 
weight  of  the  salt.  It  is  a  well-known  fact,  that  a  certain 
degree  of  heat  will  convert  water  into  vapor,  and  that  the 
same  degree  of  heat  does  not  effect  any  change  on  the  form 
of  salt.  The  mixture  of  salt  and  water  being  exposed  to 
this  temperature,  the  water  will  gradually  evaporate,  disen- 
gaging itself  from  the  salt  with  which  it  has  been  combined. 
When  so  much  of  the  water  has  evaporated,  that  what  re- 
mains is  insufficient  to  keep  in  solution  the  whole  of  the  salt, 
a  part  of  it  thus  disengaged  from  the  water  will  return  to  the 
solid  state.  The  saline  particles  will  not  in  this  case  collect 
in  irregular  solid  molecules,  but  will  exhibit  themselves  in 
particles  of  regular  figures,  terminated  by  plane  surfaces, 
the  figures  being  always  the  same  for  the  same  species  of  salt, 
but  different  for  different  species.  There  are  several  cir- 
cumstances in  the  formation  of  these  crystals  which  merit 
attention. 

If  one  of  the  crystals  be  detached  from  the  others,  and  the 
progress  of  its  formation  observed,  it  will  be  found  gradually 
to  increase,  always  preserving  its  original  figure.  Since  its 
increase  must  be  caused  by  the  continued  accession  of  saline 
particles,  disengaged  by  the  evaporation  of  the  water,  it  fol- 
lows that  these  particles  must  be  so  formed,  that,  by  attaching 
themselves  successively  to  the  crystal,  they  maintain  the  reg- 
ularity of  its  bounding  planes,  and  preserve  their  mutual  in- 
clinations unvaried. 

Suppose  a  crystal  to  be  taken  from  the  liquid  during  the 
process  of  crystallization,  and  a  piece  broken  from  it  so  as  to 


CHAP.  II.  CRYSTALLIZATION.  13 

destroy  the  regularity  of  its  form  ;  if  the  crystal  thus  broken 
be  restored  to  the  liquid,  it  will  be  observed  gradually  to 
resume  its  regular  form,  the  atoms  of  salt  successively  dis- 
missed by  the  vaporizing  water  filling  up  the  irregular  cavi- 
ties produced  by  the  fracture.  Hence  it  follows,  that  the 
saline  particles  which  compose  the  surface  of  the  crystal, 
and  those  which  form  the  interior  of  its  mass,  are  similar, 
and  exert  similar  attractions  on  the  atoms  disengaged  by  the 
water. 

All  these  details  of  the  process  of  crystallization  are  very 
evident  indications  of  a  determinate  figure  in  the  ultimate 
atoms  of  the  substances  which  are  crystallized.  But  besides 
the  substances  which  are  thus  reduced  by  art  to  the  form 
of  crystals,  there  are  larger  classes  which  naturally  exist  in 
that  state.  There  are  certain  planes,  called  planes  of  cleav- 
age, in  the  directions  of  which  natural  crystals  are  easily 
divided.  These  planes,  in  substances  of  the  same  kind, 
always  have  the  same  relative  position,  but  differ  in  different 
substances.  The  surfaces  of  the  planes  of  cleavage  are  quite 
invisible  before  the  crystal  is  divided  ;  but  when  the  parts 
are  separated,  these  surfaces  exhibit  a  most  intense  polish, 
which  no  effort  of  art  can  equal. 

We  may  conceive  crystallized  substances  to  be  regular 
mechanical  structures  formed  of  atoms  of  a  certain  figure, 
on  which  the  figure  of  the  whole  structure  must  depend.  The 
planes  of  cleavage  are  parallel  to  the  sides  of  the  constituent 
atoms  ;  and  their  directions,  therefore,  form  so  many  condi- 
tions for  the  determination  of  its  figure.  The  shape  of  the 
atoms  being  thus  determined,  it  is  not  difficult  to  assign  all 
the  various  ways  in  which  they  may  be  arranged,  so  as  to 
produce  figures  which  are  accordingly  found  to  correspond 
with  the  various  forms  of  crystals  of  the  same  substance. 

(18.)  When  these  phenomena  are  duly  considered  and 
compared,  little  doubt  can  remain  that  all  substances  suscep- 
tible of  crystallization,  consist  of  atoms  of  determinate  figure. 
This  is  the  case  with  all  solid  bodies  whatever,  which  have 
come  under  scientific  observation,  for  they  have  been  sever- 
ally found  in  or  reduced  to  a  crystallized  form.  Liquids 
crystallize  in  freezing ;  and  if  aeriform  fluids  could  by  any 
means  be  reduced  to  the  solid  form,  they  would  probably  also 
manifest  the  same  effect.  Hence  it  appears  reasonable  to 
presume,  that  all  bodies  are  composed  of  atoms ;  that  the 
different  qualities  with  which  we  find  different  substances 


14  THE  ELEMENTS  OF  MECHANIC'S.  CHAP.  II. 

endued,  depend  on  the  magnitude  and  figure  of  these  atoms; 
that  these  atoms  are  indestructible  and  immutaHo  by  any 
natural  process,  for  we  find  the  qualities  which  depend  on 
them  unchangeably  the  same  under  all  the  influences  to 
which  they  have  been  submitted  since  their  creation  ;  that 
these  atoms  are  so  minute  in  their  magnitude,  that  they  can- 
not be  observed  by  any  means  which  human  art  has  yet  con- 
trived ;  but  still  that  magnitudes  can  be  assigned  which  they 
do  not  exceed. 

It  is  proper,  however,  to  observe  here,  that  the  various 
theorems  of  mechanical  science  do  not  rest  upon  any  hypoth- 
esis concerning  these  atoms  as  a  basis.  They  are  not  infer- 
red from  this  or  any  other  supposition,  and  therefore  their 
truth  would  not  be  in  anywise  disturbed,  even  though  it 
should  be  established  that  matter  is  physically  divisible  in 
infinitum.  The  basis  of  mechanical  science  is  observed  facts ; 
and  since  the  reasoning  is  demonstrative,  the  conclusions 
have  the  same  degree  of  certainty  as  the  facts  from  which 
they  are  deduced. 

(19.)  Porosity. — The  volume  of  a  body  is  the  quantity 
of  space  included  within  its  external  surfaces.  The  mass 
of  a  body  is  the  collection  of  atoms  or  material  particles 
of  which  it  consists.  Two  atoms  or  particles  are  said  to  be 
in  contact,  when  they  have  approached  each  other  until  ar- 
rested by  their  mutual  impenetrability.  If  the  component 
particles  of  a  body  were  in  contact,  the  volume  would  be 
completely  occupied  by  the  mass.  But  this  is  not  the  case. 
We  shall  presently  prove,  that  the  component  particles  of  no 
known  substance  are  in  absolute  contact.  Hence  it  follows 
that,  the  volume  consists  partly  of  material  particles,  and 
partly  of  interstitial  spaces,  which  spaces  are  either  absolute- 
ly void  and  empty,  or  filled  by  some  substance  of  a  different 
species  from  the  body  in  question.  These  interstitial  spaces 
are  called  pores. 

In  bodies  which  are  constituted  uniformly  throughout  their 
entire  dimensions,  the  component  particles  and  the  pores  are 
uniformly  distributed  through  the  volume  ;  that  is,  a  given 
space  in  one  part  of  the  volume  will  contain  the  same  quantity 
of  matter  and  the  same  quantity  of  porqs  as  an  equal  space  in 
another  part. 

(20.)  The  proportion  of  the  quantity  of  matter  to  the 
magnitude  is  called  the  density.  Thus  if,  of  two  substances, 
one  contains  in  a  given  space  twice  as  much  matter  as  the 


CHAP.  II.  POROSITY.  15 

other,  it  is  said  to  be  "  twice  as  dense."  The  density  of  bodies 
is,  therefore,  proportionate  to  the  closeness  or  proximity  of 
their  particles ;  and  it  is  evident,  that  the  greater  the  density, 
the  less  will  be  the  porosity. 

The  pores  of  a  body  are  frequently  filled  with  another 
body  of  a  more  subtile  nature.  If  the  pores  of  a  body  on  the 
surface  of  the  earth,  and  exposed  to  the  atmosphere,  be  great- 
er than  the  atoms  of  air,  then  the  air  will  pervade  the  pores. 
This  is  found  to  be  the  case  of  many  sorts  of  wood  which 
have  open  grains.  If  a  piece  of  such  wood,  or  of  chalk, 
or  of  sugar,  be  pressed  to  the  bottom  of  a  vessel  of  water, 
the  air  which  fills  the  pores  will  be  observed  to  escape  in 
bubbles,  and  to  rise  at  the  surface,  the  water  pervading  the 
pores,  and  taking  its  place. 

If  a  tall  vessel  or  tube,  having  a.  wooden  bottom,  be  filled 
with  quicksilver,  the  liquid  metal  will  be  forced  by  its  own 
weight  through  the  pores  of  the  wood,  and  will  be  seen  es- 
caping in  a  silver  shower  from  the  bottom. 

(21.)  The  process  of  filtration,  in  the  arts,  depends  on 
the  presence  of  pores  of  such  a  magnitude  as  to  allow  a  pas- 
sage to  the  liquid,  but  to  refuse  it  to  those  impurities  from 
which  it  is  to  be  disengaged.  Various  substances  are  used 
as  filtres  ;  but,  whatever  be  used,  this  circumstance  should 
always  be  remembered,  that  no  substance  can  be  separated 
from  a  liquid  by  filtration,  except  one  whose  particles  are 
larger  than  those  of  the  liquid.  In  general,  filtres  are  used 
to  separate  solid  impurities  from  a  liquid.  The  most  ordinary 
filtres  are  soft  stone,  paper  and  charcoal. 

(22.)  All  organized  substances  in  the  animal  and  vegeta- 
ble kingdoms  are,  from  their  very  natures,  porous  in  a  high 
degree.  Minerals  are  porous  in  various  degrees.  Among 
the  siliceous  stones  is  one  called  kydropk€mettwhich  manifests 
its  porosity  in  a  very  remarkable  manner.  The  stone,  in  its 
ordinary  state,  is  semi-transparent.  If,  however,  it  be  plung- 
ed in  water,  when  it  is  withdrawn,  it  is  as  translucent  as  glass. 
The  pores,  in  this  case,  previously  filled  with  air,  are  pervaded 
by  the  water,  between  which  and  the  stone  there  subsists  a 
physical  relation,  by  which  the  one  renders  the  other  perfectly 
transparent. 

Larger  mineral  masses  exhibit  degrees  of  porosity  not  less 
striking.  Water  percolates  through  the  sides  and  roofs  of 
caverns  and  grottoes,  and,  being  impregnated  with  calcareous 


16  THE  ELEMENTS  OF  MECHANICS.  CHAP.  II. 

and  other  earths,  forms  stalactites,  or  pendent  protuberances, 
which  present  a  curious  appearance. 

(23.)  Compressibility. — That  quality,  in  virtue  of  which 
a  body  allows  its  volume  to  be  diminished  without  diminish- 
ing its  mass,  is  called  compressibility.  This  effect  is  produced 
by  bringing  the  constituent  particles  more. close  together,  and 
thereby  increasing  the  density  and  diminishing  the  pores. 
This  effect  may  be  produced  in  several  ways;  but  the  name 
"  compressibility"  is  only  applied  to  it  when  it  is  caused  by 
the  agency  of  mechanical  force,  as  by  pressure  or  percussion. 

All  known  bodies,  whatever  be  their  nature,  are  capable 
of  having  their  dimensions  reduced  without  diminishing 
their  mass  ;  and  this  is  one  of  the  most  conclusive  proofs  that 
all  bodies  are  porous,  or  that  the  constituent  atoms  are  not  in 
contact;,,  for  the  space  by  which  the  volume  maybe  diminish- 
ed must,  before  the  diminution,  consist  of  pores. 

(24.)  Some  bodies,  when  compressed  by  the  agency  of 
mechanical  force,  will  resume  their  former  dimensions  with 
a  certain  force  when  relieved  from  the  operation  of  the  force 
which  has  compressed  them.  This  property  is  called  elastici- 
ty ;  and  it  follows,  from  this  definition,  that  all  elastic  bodies 
must  be  compressible,  although  the  converse  is  not  true,  com- 
pressibility riot  necessarily  implying  elasticity. 

(25.)  Dilatability . — This  quality  is  the  opposite  of  com- 
pressibility. It  is  the  capability  observed  in  bodies  to  have 
their  volume  enlarged  without  increasing  their  mass.  This 
effect  may  be  produced  in  several  ways.  In  ordinary  circum- 
stances, a  body  may  exist  under  the  constant  action  of  a 
pressure  by  which  its  volume  and  density  are  determined.  It 
may  happen,  that,  on  the  occasional  removal  of  that  pressure, 
the  body  will  dilate  by  a  quality  inherent  in  its  constitution. 
This  is  the  case  with  common  air.  Dilatation  may  also  be 
the  effect  of  heat,  as  will  presently  appear. 

The  several  qualities  of  bodies  which  we  have  noticed  in 
this  chapter,  when  viewed  in  relation  to  each  other,  present 
many  circumstances  worthy  of  attention. 

••(26.)  It  is  a  physical  law,  to  which  there  is  no  real  ex- 
ception, that  an  increase  in  the  temperature,  or  degree  of 
heat  by  which  a  body  is  affected,  is  accompanied  by  an  in- 
crease of  volume  ;  and  that  a  diminution  of  temperature  is 
accompanied  by  a  diminution  of  volume.  The  apparent  ex- 
ceptions to  this  law  will  be  noticed  and  explained  in  our 


CHAP.  II. 


DILATABILITY.  17 


treatise  on  HEAT.  Hence  it  appears  that  the  reduction  of 
temperature  is  an  effect  which,  considered  mechanically,  is 
equivalent  to  compression  or  condensation,  since  it. diminishes 
the  volume  without  altering;  the  mass  ;  and  since  this  is  an 
effect  of  which  all  bodies  whatever  are  susceptible,  it  follows 
that  all  bodies  whatever  have  pores.  (23.) 

The  fact,  that  the  elevation  of  temperature  produces  an 
increase  of  volume,  is  manifested  by  numerous  experiments. 

(•37.)  If  a  flaccid  bladder  be  tied  at  the  mouth,  so  as 
to  stop  the  passage  of  air,  and  be  then  held  before  a  fire, 
it  will  gradually  swell,  and  assume  the  appearance  of  being 
fully  inflated.  The  small  quantity  of  air  contained  in  the 
bladder  is,  in  this  case,  so  much  dilated  by  the  heat,  that  it 
occupies  a  considerably  increased  space,  and  fills  the  bladder, 
of  which  it  before -only  occupied  a  small  part.  When  the 
I  ladder  is  removed  from  the  fire,  and  allowed  to  resume  its 
former  temperature,  the  air  returns  to  its  former  dimensions, 
and  the  bladder  becomes  again  flaccid. 

(28.)  Let  A  B,  fig.  2.  be  a  glass  tube,  with  a  bulb  at  the 
end  A  ;  and  let  the  bulb  A,  and  a  part  of  the  tube,  be  filled 
with  any  liquid,  colored  so  as  to  be  visible.  Let  C  be  the 
level  of  the  liquid  in  the  tube.  If  the  bulb  be  now  exposed 
to  heat,  by  being  plunged  in  hot  water,  the  level  of  the  liquid 
C  will  rapidly  rise  towards  B.  This  effect  is  produced  by  the 
dilatation  of  the  liquid  in  the  bulb,  which  filling  a  greater 
space,  a  part  of  it  is  forced  into  the  tube.  This  experiment 
may  easily  be  made  with  a  common  glass  tube  and  a  little 
port  wine. 

Thermometers  are  constructed  on  this  principle,  the  rise 
of  the  liquid  in  the  tube  being  used  as  an  indication  of  the 
degree  of  heat  which  causes  it.  A  particular  account  of 
these  useful  instruments  will  be  found  in  our  treatise  on  HEAT. 

(29.)  The  change  , of  dimension  of  solids  produced  by 
changes  of  temperature,  being  much  less  than  that  of  bodies 
in  the  liquid  or  aeriform  state,  is  not  so  easily  observable. 
A  remarkable  instance  occurs  in  the  process  of  shoeing  the 
wheels  of  carriages.  The  rim  of  iron  with  which  the  wheel 
is  to  be  bound,  is  made,  in  the  first  instance,  of  a  diameter 
somewhat  less  than  that  of  the  wheel ;  but,  being  raised 
by  the  application  of  fire  to  a  very  high  temperature,  its 
volume  receives  such  an  increase,  that  it  will  be  sufficient 
to  embrace  and  surround  the  wheel.  When  placed  upon  the 
wheel,  it  is  cooled,  and,  suddenly  contracting  its  dimensions, 
2*  ' 


18  THE  ELEMENTS  OF  MECHANIC'S.  CHAP.  11. 

binds  the  parts  of  the  wheel  firmly  together,  and  becomes 
securely  seated  in  its  place  upon  the  face  of  the  fellies. 

(30.)  It.  frequently  happens  that  the  stopper  of  a  glass 
bottle  or  decanter  becomes  fixed  in  its  place  so  firmly,  that 
the  exertion  of  force  sufficient  to  withdraw  it  would  endanger 
the  vessel.  In  this  case,  if  a  cloth  wetted  with  hot  water 
be  applied  to  the  neck  of  the  bottle,  the  glass  will  expand, 
and  the  neck  will  be  enlarged,  so  as  to  allow  the  stopper 
to  be  easily  withdrawn. 

(31.)  The  contraction  of  metal  consequent  upon  change 
of  temperature  has  been  applied  some  time  ago  in  Paris 
to  restore  the  walls  of  a  tottering  building  to  their  proper 
position.  In  the  Conservatoire,  dts  Arts  ct  Meticrcs,  the  walls 
of  a  part  of  the  building  were  forced  out  of  the  perpendicular 
by  the  weight  of  the  roof,  so  that  each  wall  was  leaning  out- 
wards. M.  Molard  conceived  the  notion  of  applying  the  irre- 
sistible force  with  which  metals  contract  in  cooling,  to  draw 
the  walls  together.  Bars  of  iron  were  placed  in  parallel 
directions  across  the  building,  and  at  right  angles  to  the  di- 
rection of  the  walls.  Being  passed  through  the  walls,  nuts 
were  screwed  on  their  ends,  outside  the  building.  Every 
alternate  bar  was  then  heated  by  lamps,  and  the  nuts  screw- 
ed close  to  the  walls.  The  bars  were  then  cooled,  and  the 
lengths  being  diminished  by  contraction,  the  nuts  on  their 
extremities  were  drawn  together,  and  with  them  the  walls 
were  drawn  through  an  equal  space.  The  same  process  was 
repeated  with  the  intermediate  bars,  and  soon  alternately  until 
the  walls  were  brought  into  a  perpendicular  position. 

(32  )  Since  there  is  a  continual  change  of  temperature 
in  all  bodies  on  the  surface  of  the  globe,  it  followrs,  that  there 
is  also  a  continual  change  of  magnitude.  The  substances 
which  surround  us  are  constantly  swelling  and  contracting 
under  the  vicissitudes  of  heat  and  cold.  They  grow  smaller 
in  winter,  and  dilate  in  summer.  They  swell  their  bulk 
on  a  warm  day,  and  contract  it  on  a  cold  one.  These  curi- 
ous phenomena  are  not  noticed,  only  because  our  ordinary 
means  of  observation  are  not  sufficiently  accurate  to  appre- 
ciate them.  Nevertheless,  in  some  familiar  instances  the 
effect  is  very  obvious.  In  warm  weather,  the  flesh  swells, 
the  vessels  appear  filled,  the  hand  is  plump,  and  the  skin 
distended.  In  cold  weather,  when  the  body  has  been  expos- 
ed to  the  open  air,  the  flesh  appears  to  contract,  the  vessels 
shrink,  and  the  skin  shrivels. 


CHAP.  II. 


COMPRESSIBILITY.  19 


(33.)  The  phenomena  attending  change  of  temperature 
are  conclusive  proofs  of  the  universal  porosity  of  material  sub- 
stances, hut  they  are  not  the  only  proofs.  Many  substances 
admit  of  compression  by  the  mere  agency  of  mechanical 
force. 

Let  a  small  piece  of  cork  be  placed  floating  on  the  surface 
of  water  in  a  basin  or  other  vessel,  and  an  empty  glass  goblet 
be  inverted  over  the  cork,  so  that  its  edge  just  meets  the 
water.  A  portion  of  air  will  then  be  confined  in  the  goblet, 
and  detached  from  the  remainder  of  the  atmosphere.  If  the 
goblet  be  now  pressed  downwards,  so  as  to  be  entirely  im- 
mersed, it  will  be  observed,  that,  the  water  will  not  fill  it, 
being  excluded  by  the  impenetrability  of  the  air  enclosed  in 
it.  This  experiment,  therefore,  is  decisive  of  the  fact,  that 
air,  one  of  the  most  subtile  and  attenuated  substances  we 
know  of,  possesses  the  quality  of  impenetrability.  It  abso- 
lutely excludes  any  other  body  from  the  space  which  it  occu- 
pies at  any  given  moment. 

But  although  the  water  does  not  fill  the  goblet,  yet  if  the 
position  of  the  cork  which  floats  upon  its  surface  be  noticed, 
it  will  be  found  that  the  level  of  the  water  within  has  risen 
above  its  edge  or  rim.  In  fact,  the  water  has  partially  filled 
the  goblet,  and  the  air  has  been  forced  to  contract  its  dimen- 
sions. This  effect  is  produced  by  the  pressure  of  the  incum- 
bent water  forcing  the  surface  in  the  goblet  against  the  air, 
which  yields  until  it  is  so  far  compressed  that  it  acquires  a 
force  able  to  withstand  this  pressure.  Thus  it  appears  that 
air  is  capable  of  being  reduced  in  its  dimensions  by  mechani- 
cal pressure,  independently  of  the  agency  of  heat.  It  is 
compressible. 

That  this  effect  is  the  consequence  of  the  pressure  of  the 
liquid,  will  be  easily  made  manifest  by  showing  that,  as  the 
pressure  is  increased,  the  air  is  proportionally  contracted  in 
its  dimensions  ;  and  as  it  is  diminished,  the  dimensions  are, 
on  the  other  hand,  enlarged.  If  the  depth  of  the  goblet 
in  the  water  be  increased,  the  cork  will  be  seen  to  rise  in 
it,  showing  that  the  increased  pressure,  at  the  greater  depth, 
causes  the  air  in  the  goblet  to  be  more  condensed.  If, 
on  the  other  hand,  the  goblet  be  raised  toward  the  sur- 
face, the  cork  will  be  observed  to  descend  toward  the  edge, 
showing  that  as  it  is  relieved  from  the  pressure  of  the 
liquid,  the  air  gradually  approaches  to  its  primitive  dimen- 
sions. 


20  THE  ELEMENTS  OF  MECHANICS.  CHAP.  II. 

(34.)  These  phenomena  also  prove,  that  air  has  the  prop- 
erty of  elasticity.  If  it  wore  simply  compressible,  and  not 
elastic,  it  would  retain  the  dimensions  to  which  it  was  reduced 
by  the  pressure  of  the  liquid  ;  but  this  is  not  found  to  be 
the  result.  As  the  compressing  force  is  diminished,  so  in 
the  same  proportion  does  the  air,  by  its  elastic  virtue,  exert  a 
force  by  which  it  resumes  its  former  dimensions. 

That  it  is  the  air  alone  which  excludes  the  water  from  the 
goblet,  in  the  preceding  experiments,  can  easily  be  proved. 
When  the  goblet  is  sunk  deep  in  the  vessel  of  water,  let  it  be 
inclined  a  little  to  one  side  until  its  mouth  is  presented  to- 
wards the  side  of  the  vessel ;  let  this  inclination  be  so  regu- 
lated, that  the  surface  of  the  water  in  the  goblet  shall  just  reach 
its  edge.  Upon  a  slight  increase  of  inclination,  air  will  be 
observed  to  escape  from  the  goblet,  and  to  rise  in  bubbles 
to  the  surface  of  the  water.  If  the  goblet  be  then  restored 
to  its  position,  it  will  be  found  that  the  cork  will  rise  higher 
in  it  than  before  the  escape  of  the  air.  The  water  in  this 
case  rises,  and  fills  the'space  which  the  air  allowed  to  escape 
has  deserted.  The  same  process  may  be  repeated  until  all  the 
air  has  escaped,  and  the!)  the  goblet  will  be  completely  filled 
by  the  water. 

(35.)  Liquids  are  compressible  by  mechanical  force  in  so 
slight  a  degree,  that  they  are  considered,  in  all  hydrostatical 
treatises,  as  incompressible  fluids.  They  are,  however,  not 
absolutely  incompressible,  but  yield  slightly  to  very  intense 
pressure.  The  question  of  the  compressibility  of  liquids  was 
raised  at  a  remote  period  in  the  history  of  science.  Nearly 
two  centuries  ago,  an  experiment  was  instituted  at  the  Acad- 
emy del  Oimcnto  in  Florence,  to  ascertain  whether  water  be 
compressible.  With  this  view,  a  hollow  ball  of  gold  was 
filled  with  the  liquid,  and  the  aperture  exactly  and  firmly 
closed.  The  globe  was  then  submitted  to  a  very  severe 
pressure,  by  which  its  figure  was  slightly  changed.  Now 
it  is  proved  in  geometry,  that  a  globe  has  this  peculiar  prop- 
erty, that  any  change  whatever  in  its  figure  must  necessarily 
diminish  its  volume  or  contents.  Hence  it  was  inferred,  that 
if  the  water  did  not  issue  through  the  pores  of  the  gold,  or 
burst  the  globe,  its  compressibility  would  be  established.  The 
result  of  the  experiment  was,  that  the  water  did  ooze  through 
the  pores,  and  covered  the  surface  of  the  globe,  presenting 
the  appearance  of  dew,  or  of  steam  cooled  by  the  metal.  But 
this  experiment  was  inconclusive.  It  is  quite  true,  that  if 


CHAP!  II.  COMPRESSIBILITY ELASTICITY.  21 

the  water  had  not  escaped  upon  the  change  of  figure  of  the 
globe,  the  compressibility  of  the  liquid  would  have  been  estab- 
lished. The  escape  of  the  water  does  not,  however,  prove 
its •  incompregsibility.  To  accomplish  this,  it  would  be  neces- 
sary first  to  measure  accurately  the  volume  of  water  which 
transuded  by  compression,  and  next  to  measure  the  diminu- 
tion of  volume  which  the  vessel  suffered  by  its  change  of 
figure.  If  this  diminution  were  greater  than  the  volume  of 
water  which  escaped,  it  would  follow  that  the  water  remain- 
ing in  the  globe  had  been  compressed,  notwithstanding  the 
escape  of  the  remainder.  But  this  could  never  be  accom- 
plished with  the  delicacy  and  exactitude  necessary  in  such  an 
experiment ;  and,  consequently,  as  far  as  the  question  of  the 
compressibility  of  water  was  concerned,  nothing  was  proved. 
It  forms,  however,  a  very  striking  illustration  of  the  porosity 
of  so  dense  a  substance  as  gold,  and  proves  that  its  pores  are 
larger  than  the  elementary  particles  of  water,  since  they  are 
capable  of  passing  through  them. 

(30.)  It  has  since  been  proved,  that  water,  and  other 
liquids,  are  compressible.  In  the  year  1761,  Canton  com- 
municated to  the  Royal  Society  the  results  of  some  experi- 
ments which  proved  this  fact.  He  provided  a  glass  tube 
with  a  bulb,  such  as  that  described  in  (28.),  and  filled  the 
bulb  and  a  part  of  the  tube  with  the  liquid  well  purified 
from  air.  He  then  placed  this  in  an  apparatus  called  a  con- 
denser, by  which  he  was  enabled  to  submit  the  surface  of 
the  liquid  in  the  tube  to  very  intense  pressure  of  condensed 
air.  lie  found  that  the  level  of  the  liquid  in  the  tube  fell  in 
a  very  perceptible  degree  upon  the  application  of  the  pres- 
sure. The  same  experiment  established  the  fact,  that  liquids 
are  clastic. ;  for,  upon  removing  the  pressure,  the  liquid  rose 
to  its  original  level,  and  therefore  resumed  its  former  dimen- 
sions. 

(37.)  Elasticity  does  not  always  accompany  compressibil- 
ity. If  lead  or  iron  be  submitted  to  the  hammer,  it  may  be 
hardened  and  diminished-  in  its  volume  ;  but  it  will  not  re- 
sume its  former  volume  after  each  stroke  of  the  hammer. 

(3.^.)  There  are  some  bodies  which  maintain  the  state 
of  density  in  which  they  are  commonly  found  by  the  continual 
agency  of  mechanical  pressure  ;  and  such  bodies  are  endued 
with  a  quality,  in  virtue  of  which  they  would  enlarge  their 
dimensions  without  limit,  if  the  pressure  which  confines  them 
were  removed.  Such  bodies  are  called  clastic  fluids  or  gases 


22  THE  ELEMENTS  OF  MECHANICS.        CHAP.  II. 

and  always  exist  in  the  form  of  common  air,  in  whose  me- 
chanical properties  they  participate.  They  are  hence  often 
called  aeriform  fluids, 

Those  who  are  provided  with  an  air-pump  can  easily  estab- 
lish this  property  experimentally.  Take  a  flaccid  bladder, 
such  as  that  already  described  in  (27.),  and  place  it  under 
the  glass  receiver  of  an  air-pump  :  by  this  instrument  we 
shall  be  able  to  remove  the  air  which  surrounds  the  bladder 
under  the  receiver,  so  as  to  relieve  the  small  quantity  of  air 
which  is  enclosed  in  the  bladder  from  the  pressure  of  the 
external  air  :  when  this  is  accomplished,  the  bladder  will  be 
observed  to  swell,  as  if  it  were  inflated,  and  will  be  perfectly 
distended.  The  air  contained  in  it,  therefore,  has  a  tendency 
to  dilate,  which  takes  effect  when  it  ceases  to  be  resisted  by 
the  pressure  of  surrounding  air. 

(39.)  It  has  been  stated  that  the  increase  or  diminution 
of  temperature  is  accompanied  by  an  increase  or  diminution 
of  volume.  Related  to  this,  there  is  another  phenomenon 
too  remarkable  to  pass  unnoticed,  although  this  is  not  the 
proper  place  to  dwell  upon  it:  it  is  the  con  verse  of  the  former; 
viz.  that  an  increase  or  diminution  of  bulk  is  accompanied 
by  a  diminution  or  increase  of  temperature.  As  the  applica- 
tion of  heat  from  some  foreign  source  produces  an  increase 
of  dimensions,  so  if  the  dimensions  be  increased  from  any 
other  cause,  a  corresponding  portion  of  the  hea^  which  the 
body  had  before  the  enlargement,  will  be  absorbed  in  the 
process,  and  the  temperature  will  be  thereby  diminished. 
In  the  same  way,  since  the  abstraction  of  heat  causes  a  dim- 
inution of  volume,  so  if  that  diminution  be  caused  by  any 
other  means,  the  body  will  ff-for.  out,  the  heat  which  in  the 
other  case  was  abstracted,  and  will  rise  in  its  temperature. 

Numerous  and  well-known  facts  illustrate  these  observa- 
tions. A  smith,  by  hammering  a  piece  of  bar  iron,  and  there- 
by compressing  it,  will  render  it  re.d  hot.  When  air  is  vio- 
lently compressed,  it  becomes  so  hot  as  to  ignite  cotton  and 
other  substances.  An  ingenious  instrument  for  producing 
a  light  for  domestic  uses  has  been  constructed,  consisting  of  a 
small  cylinder,  in  which-  a  solid  piston  moves  air-tight :  a  little 
tinder,  or  dry  sponge,  is  attached  to  the  bottom  of  the  piston, 
which  is  then  violently  forced  into  the  cylinder  :  the  air  be- 
tween the  bottom  of  the  cylinder  and  the  piston  becomes 
intensely  compressed,  and  evolves  so  much  heat  as  to  light 
the  tinder. 


CHAP.  III.  '    INERTIA.  23 

In  all  the  cases  where  friction  or  percussion  produces  heat 
or  fire,  it  is  because  they  are  means  of  compression.  The 
effects  of  flints,  of  pieces  of  wood  rubbed  together,  the 
warmth  produced  by  friction  on  the  flesh,  are  all  to  be 
attributed  to  the  same  cause. 


CHAPTER  III. 

INERTIA. 

(40.)  THE  quality  of  matter  which  is  of  all  others  the  most 
important  in  mechanical  investigations,  is  that  which  has  been 
called  Inertia. 

Matter  is  incapable  of  spontaneous  change.  This  is  one 
of  the  earliest  and  most  universal  results  of  human  observa- 
tion ;  it  is  equivalent  to  stating  that  mere  matter  is  deprived 
of  life  ;  for  spontaneous  action  is  the  only  test  of  the  pres- 
ence of  the  living  principle.  If  we  see  a  mass  of  matter 
undergo  any  change,  we  never  seek  for  the  cause  of  that 
change  in  the  body  itself;  we  look  for  some  external  cause 
producing  it.  This  inability  for  voluntary  change  of  state  or 
qualities  is  a  more  general  principle  than  inertia.  At  any 
given  moment  of  time,  a  body  must  be  in  one  or  other  of  two 
states,  rest  or  motion.  Inertia,  or  inactivity,  signifies  the 
total  absence  of  power  to  change  this  state.  A  body  endued 
with  inertia  cannot  of  itself,  and  independent  of  all  external 
influence,  commence  to  move  from  a  state  of  rest ;  neither 
can  it,  when  moving,  arrest  its  progress,  and  become  quies- 
cent. 

(41.)  The  same  property  by  which  a  body  is  unable  by 
any  power  of  its  own  to  pass  from  a  state  of  rest  to  one  of 
motion,  or  vice  versa ,  also  renders  it  incapable  of  increasing 
or  diminishing  any  motion  which  it  may  have  received  from 
an  external  cause.  If  a  body  be  moving  in  a  certain  direc 
tion  at  the  rate  of  ten  miles  per  hour,  it  cannot,  by  any  ener- 
gy of  its  own,  change  its  rate  of  motion  to  eleven  or  nine 
miles  an  hour.  This  is  a  direct  consequence  of  that  mani- 
festation of  inertia  which  has  just  been  explained.  For  the 
same  power  which  would  cause  a  body  moving  at  ten  miles  an 
hour  to  increase  its  rate  to  eleven  miles,  would  also  cause  the 
same  body  at  rest  to  commence  moving  at  the  rate  of  one 


24  THE  ELEMENTS  OF  MECHANICS.       CHAP.  III. 

mile  an  hour  ;  and  the  same  power  which  would  cause  a 
body  moving  at  the  rate  of  ten  miles  an  hour  to  wove  at  the 
rate  of  nine  miles  in  the  hour,  would  cause  the  :-ame  body 
moving  at  the  rate  of  one  mile  an  hour  to  become  quiescent. 
It  therefore  appears,  that  to  increase  or  diminish  the  motion 
of  a  body  is  an  effect  of  the  same  kind  as  to  change  the  state 
of  rest  into  that  of  motion,  or  vice  versa. 

(42.)  The  effects  and  phenomena  which  hourly  fill  under 
our  observation  afford  unnumbered  examples  of  the  inabil- 
ity of  lifeless  matter  to  put  itself  into  motion,  or  to  increase 
any  motion  which  may  have  been  communicated  to  it.  But 
it  does  not  happen  that  we  have  the  same  direct  and  frequent 
evidence  of  its  inability  to  destroy  or  diminish  any  motion 
which  it  may  have  received.  And  hence  it  arises,  that,  while 
no  one  will  deny  to  matter  the  former  effect  of  inertia,  few 
will  at  firs;  acknowledge  the  latter.  Indeed,  even  so  late 
as  the  time  of  KEPLEU,  philosophers  themselves  held  it  as  a 
maxim,  that  "  matter  is  more  inclined  to  rest  than  to  motion ;" 
we  ought  not,  therefore,  to  be  surprised  if,  in  the  present  day, 
those  who  have  not  been  conversant  with  physical  science 
are  slow  to  believe  that  a  body  once  put  in  motion  would 
continue  for  ever  to  move  with  the  same  velocity,  if  it  were 
not  stopped  by  some  external  cause. 

Reason,  assisted  by  observation,  will,  however,  soon  dispel 
this  illusion.  Experience  shows  us  in  various  ways,  that  the 
same  causes  which  destroy  motion  in  one  direction  are  capable 
of  producing  as  much  motion  in  the  opposite  direction.  Thus, 
if  a  wheel,  spinning  on  its  axis  with  a  certain  velocity,  be 
stopped  by  a  hand  seizing  one  of  the  spokes,  the  effort  which 
accomplishes  this  is  exactly  the  same  as,  had  the  wheel  been 
previously  at  rest,  would  have  put  it  in  motion  in  the  opposite 
direction  with  the  same  velocity.  If  a  carriage  drawn  by 
horses  be  in  motion,  the  saint'  exertion  of  power  in  the  horses 
is  necessary  to  stop  it,  as  would  be  necessary  to  back  it,  if  it 
were  at  rest.  Now,  if  this  be  admitted  as  a  general  principle, 
it  must  be  evident  that  a  body  which  can  destroy  or  diminish 
its  own  motion  must  also  be  capable  of  putting  itself  into 
motion  from  a  state  of  rest,  or  of  increasing  any  motion 
which  it  has  received.  But  this  latter  is  contrary  to  all 
experience,  and  therefore  we  are  compelled  to  admit  that 
a  body  cannot  diminish  or  destroy  any  motion  which  it  has 
received. 

Let  us  inquire  why  we  are  more  disposed  to  admit  the 


CHAP.  III.  INERTIA.  25 

inability  of  matter  to  produce  than  to  destroy  motion  in  itself. 
We  see  most  of  those  motions  which  take  place  around  us  on 
the  surface  of  the  earth  subject  to  gradual  decay,  and  if  not 
renewed  from  time  to  time,  they  at  length  cease.  A  stone 
rolled  along  the  ground,  a  wheel  revolving  on  its  axis,  the 
heaving  of  the  deep  after  a  storm,  and  all  other  motions  pro- 
duced in  bodies  by  external  causes,  decay,  when  the  exciting 
cause  is  suspended  ;  and  if  that  cause  do  not  renew  its  ac- 
tion, they  ultimately  cease. 

But  is  there  no  exciting  cause,  on  the  other  hand,  which 
thus  gradually  deprives  those  bodies  of  their  motion  ? — and 
if  that  cause  were  removed,  or  its  intensity  diminished, 
would  not  the  motion  continue,  or  be  more  slowly  retarded  ? 
When  a  stone  is  rolled  along  the  ground,  the  inequalities 
of  its  shape,  as  well  as  those  of  the  ground,  are  impediments 
which  retard  and  soon  destroy  its  motion.  Render  the  stone 
round,  and  the  ground  level,  and  the  motion  will  be  consider- 
ably prolonged.  But  still  small  asperities  will  remain  on  the 
stone,  and  on  the  surface  over  which  it  rolls :  substitute  for  it 
a  ball  of  highly  polished  steel,  moving  on  a  highly  polished 
steel  plane,  truly  level,  and  the  motion  will  continue  without 
sensible  diminution  for  a  very  long  period  ;  but  even  here, 
and  in  every  instance  of  motions  produced  by  art,  minute 
asperities  must  exist  on  the  surfaces  which  move  in  contact 
with  each  other,  which  must  resist,  gradually  diminish,  and 
ultimately  destroy  the  motion. 

Independently  of  the  obstructions  to  the  continuation  of 
motion  arising  from  friction,  there  is  another  impediment  to 
which  all  motions  on  the  surface  of  the  earth  are  liable — 
the  resistance  of  the  air.  How  much  this  may  affect  the 
continuation  of  motion,  appears  by  many  familiar  effects.  On 
a  calm  day,  carry  an  open  umbrella  with  its  concave  side 
presented  in  the  direction  in  which  you  are  moving,  and 
a  powerful  resistance  will  be  opposed  to  your  progress,  which 
will  increase  with  every  increase  of  the  speed  with  which  you 
move. 

We  are  not,  however,  without  direct  experience  to  prove, 
that  motions  when  unresisted  will  for  ever  continue.  In 
the  heavens  we  find  an  apparatus,  whicli  furnishes  a  sublime 
verification  of  this  principle.  There,  removed  from  all  casual 
obstructions  and  resistances,  the  vast  bodies  of  the  universe 
roll  on  in  their  appointed  paths  with  unerring  regularity, 
preserving  without  diminution  all  ftiat  motion  which  they 
3 


26  THE  ELEMENTS  OF  MECHANICS.  CHAP.  Ill 

received  at  their  creation  from  the  hand  which  launched 
them  into  space.  This  alone,  unsupported  by  other  reasons, 
would  be  sufficient  to  establish  the  quality  of  inertia ;  but 
viewed  in  connection  with  the  other  circumstances  previously 
mentioned,  no  doubt  can  remain  that  this  is  an  universal  law 
of  nature. 

(43.)  Organized  bodies  endued  with  the  living  principle, 
seem  to  be  the  only  exceptions  to  this  law.  But  even  in  these 
their  members  and  all  their  component  parts,  separately  con- 
sidered, are  inert,  and  are  subject  to  the  same  laws  as  all 
other  forms  of  matter.  The  quality  of  animation,  from  which 
they  derive  the  power  of  spontaneous  action  or  voluntary 
motion,  does  not  belong  to  the  parts,  but  to  the  whole,  and 
not  to  the  whole  by  any  obvious  or  necessary  connection, 
because  it  is  absent  in  sleep,  and  totally  removed  by  death, 
even  while  the  organization  of  every  part  remains,  to  all  ap- 
pearance, without  derangement.  Seeing,  then,  the  whole 
visible  material  universe  partaking  in  the  common  quality 
of  inertia,  unable  to  trace  the  conditions  of  life  to  any  ma- 
terial phenomena,  it  is  impossible  not  to  conclude  that  the 
will  of  animated  beings  is  the  result  of  an  immaterial  prin- 
ciple, which,  during  the  period  of  life,  governs  their  organ- 
ized bodies.  In  what  this  principle  consists,  what  is  its  seat, 
or  by  what  modes  of  action  it  moves  the  body,  we  are  wholly 
unable  to  decide.  But  the  same  principle,  analogy,  which 
guides  our  investigations  in  every  other  part  of  physical  sci- 
ence, ought  to  govern  us  in  this  ;  and  by  that  principle,  the 
spontaneous  motion  found  in  animated  beings,  but  which  in 
no  instance  is  manifested  by  mere  matter,  must  be  attrib- 
uted not  to  the  matter  which  composes  the  bodily  forms  of 
these  beings,  but  to  something  of  altogether  a  different  na- 
ture. 

Independently  of  this,  which  may  be  considered  as  the 
reasoning  proper  to  physical  science,  philosophers  have  given 
another  reason  for  assigning  animation  to  an  immaterial 
principle.  The  will,  from  the  very  nature  of  its  acts,  must 
belong  to  a  simple,  uncompounded,  and  indivisible  being, 
and  consequently  can  never  be  an  attribute  of  a  thing  which 
in  its  essence  is  the  very  reverse  of  this. 

(44.)  It  has  been  proved,  that  an  inability  to  change  the 
quantity  of  motion  is  a  consequence  of  inertia.  The  inability 
to  change  the  direction  of  motion  is  another  consequence  of 
this  quality.  The  same  cause  which  increases  or  diminishes 


CHAP.  III.          SPONTANEOUS  MOTION.  27 

motion,  would  also  give  motion  to  a  body  at  rest ;  and  there- 
fore we  inferred  that  the  same  inability  which  prevents  a 
body  from  moving  itself,  will  also  prevent  it  from  increasing 
or  diminishing  any  motion  which  it  has  received.  In  the 
same  manner  we  can  show,  that  any  cause  which  changes 
the  direction  of  motion  would  also  give  motion  to  a  body 
at  rest ;  and  therefore  if  a  body  change  the  direction  of  its 
own  motion,  the  same  body  might  move  itself  from  a  state 
of  rest ;  and  therefore  the  power  of  changing  the  direction 
of  any  motion  which  it  may  have  received  is  inconsistent 
with  the  quality  of  inertia. 

(45.)  If  a  body,  moving  from  A,  Jig.  3.  to  B,  receive  at 
B  a  blow  in  the  direction  C  B  E,  it  will  immediately  change 
its  direction  to  that  of  another  line  B  D.  The  cause  which 
produces  this  change  of  direction  would  have  put  the  body  in 
motion  in  the  direction  B  E,  had  it  been  quiescent  at  B  when 
it  sustained  the  blow. 

(46.)  Again,  suppose  G  H  to  be  a  hard  plane  surface; 
and  let  the  body  be  supposed  to  be  perfectly  inelastic.  When 
it  strikes  the  surface  at  B,  it  will  commence  to  move  along 
it  in  the  direction  B  H.  This  change  of  direction  is  pro- 
duced by  the  resistance  of  the  surface.  If  the  body,  instead 
of  meeting  the  surface  in  the  direction  A  B,  had  moved  in 
the  direction  E  B,  perpendicular  to  it,  all  motion  would  have 
been  destroyed,  and  the  body  reduced  to  a  state  of  rest. 

(47.)  By  the  former  example  it  appears,  that  the  deflecting 
cause  would  have  put  a  quiescent  body  in  motion,  and  by  the 
latter  it  would  have  reduced  a  moving  body  to  a  state  of  rest. 
Hence  the  phenomenon  of  a  change  of  direction  is  to  be 
referred  to  the  same  class  as  the  change  from  rest  to  motion, 
or  from  motion  to  rest.  The  quality  of  inertia  is,  therefore, 
inconsistent  with  any  change  in  the  direction  of  motion  which 
does  not  arise  from  an  external  cause. 

(48.)  From  all  that  has  been  here  stated,  we  may  infer 
generally,  that  an  inanimate  parcel  of  matter  is  incapable 
of  changing  its  state  of  rest  or  motion  ;  that,  in  whatever  state 
it  be,  in  that  state  it  must  for  ever  persevere,  unless  disturbed 
by  some  external  cause  ;  that  if  it  be  in  motion,  that  motion 
must  always  be  uniform^  or  must  proceed  at  the  same  rate, 
the  equal  spaces  being  moved  over  in  the  same  time  ;  any 
increase  of  its  rate  must  betray  «ome  impelling  cause,  any 
diminution  must  proceed  from  an  impeding  cause,  and  nei- 
ther of  these  causes  can  exist  in  the  body  itself;  that  such 


28  THE  ELEMENTS  OF  MECHANICS.  CHAP.  III. 

motion  must  not  only  be  constantly  of  the  same  uniform  rate, 
but  also  must  be  always  in  the  same  direction,  any  deflec- 
tion from  its  course  necessarily  arising  from  some  external 
influence. 

The  language  sometimes  used  to  explain  the  property  of 
inertia  in  popular  works,  is  eminently  calculated  to  mislead 
the  student.  The  terms  resistance  and  stubbornness  to  move 
are  faulty  in  this  respect.  Inertia  implies  absolute  passive- 
ness,  a  perfect  indifference  to  re&t  or  motion.  It  implies 
as  strongly  the  absence  of  all  resistance  to  the  reception  of 
motion,  as  it  does  the  absence  of  all  power  to  move  itself. 
The  term  vis  inertia,  or  force  of  inaitivity,  so  frequently 
used  even  by  authors  pretending  to  scientific  accuracy,  is 
still  more  reprehensible.  It  is  a  contradiction  in  terms ;  the 
term  inactivity  implying  the  absence  of  all  force. 

(49.)  Before  we  close  this  chapter,  it  may  be  advantageous 
to  point  out  gome  practical  and  familiar  examples  of  the 
general  law  of  inertia.  The  student  must,  however,  recollect, 
that  the  great  object  of  science  is  generalization,  and  that 
his  mind  is  to  be  elevated  to  the  contemplation  of  the  laws 
of  nature,  and  to  receive  a  habit  the  very  reverse  of  that 
which  disposes  us  to  enjoy  the  descent  from  generals  to  par- 
ticulars. Instances,  taken  from  the  occurrences  of  ordinary 
life,  may,  however,  be  useful  in  verifying  the  general  law, 
and  in  impressing  it  upon  the  memory  ;  and,  for  this  reason, 
we  shall  occasionally,  in  the  present  treatise,  refer  to  such 
examples  ;  always,  however,  keeping  them  in  subservience 
to  the  general  principles  of  which  they  are  manifestations, 
and  on  which  the  attention  of  the  student  should  be  fixed. 

(50.)  If  a  carriage,  a  horse,  or  a  boat,  moving  with  speed, 
be  suddenly  retarded  or  stopped,  by  any  cause  which  does 
not  at  the  same  time  affect  passengers,  riders,  or  any  loose 
bodies  which  are  carried,  they  will  be  precipitated  in  the 
direction  of  the  motion  ;  because,  by  reason  of  their  inertia, 
they  persevere  in  the  motion  which  they  shared  in  common 
with  that  which  transported  them,  and  are  not  deprived  of 
that  motion  by  the  same  cause. 

(51.)  If  a  passenger  leap  from  a  carriage  in  rapid  motion, 
he  will  fall  in  the  direction  in  which  the  carriage  is  moving 
at  the  moment  his  feet  meet  the  ground  ;  because  his  body, 
on  quitting  the  vehicle,  retains,  by  its  inertia,  the  motion 
which  it  had  in  common  with  it.  When  he  reaches  ti.e 


CHAP.  IV.  ACTION    AND    REACTION.  29 

ground,  this  motion  is  destroyed  by  the  resistance  of  the 
ground  to  the  feet,  but  is  retained  in  the  upper  and  heavier 
part  of  the  body ;  so  that  the  same  effect  is  produced  as  if 
the  feet  had  been  tripped. 

(52.)  When  a  carriage  is  once  put  in  motion  with  a  deter- 
minate speed  on  a  level  road,  the  only  force  necessary  to 
sustain  the  motion  is  that  which  is  sufficient  to  overcome  the 
friction  of  the  road ;  but  at  starting  a  greater  expenditure  of 
force  is  necessary,  inasmuch  as  not  only  the  friction  is  to  be 
overcome,  but  the  force  with  which  the  vehicle  is  intended  to 
move  must  be  communicated  to  it.  Hence  we  see  that 
horses  make  a  much  greater  exertion  at  starting  than  subse- 
quently, when  the  carriage  is  in  motion ;  and  we  may  also 
infer  the  inexpediency  of  attempting  to  start  at  full  speed, 
especially  with  heavy  carriages. 

(53.)  Coursing  owes  all  its  interest  to  the  instinctive  con- 
sciousness of  the  nature  of  inertia  which  seems  to  govern 
the  measures  of  the  hare.  The  greyhound  is  a  comparatively 
heavy  body  moving  at  the  same  or  greater  speed  in  pursuit 
The  hare  doublfs,  that  is,  suddenly  changes  the  direction  of 
her  course,  and  turns  back  at  an  oblique  angle  with  the  di- 
rection in  which  she  had  been  running.  The  greyhound, 
unable  to  resist  the  tendency  of  itjs  body  v>  persevere  in  the 
rapid  motion  it  had  acquired,  is  urged  forward  many  yards 
before  it  is  able  to  check  its  speed  and  return  to  the  pursuit. 
Meanwhile  the  hare  is  gaining  ground  in  the  other  direction, 
so  that  the  animals  are  at  a  ^very  considerable  distance  asun- 
der when  the  pursuit  is  recommenced.  In  this  way,  a  hare, 
.hough  much  less  fleet  than  a  greyhound,  will  often  escape  it. 

In  racing,  the  horses  shoot  far  beyond  the  winning-post 
Before  their  course  can  be  arrested 


CHAPTER  IV. 

ACTION    AND    REACTION 

(54.)  THE  effects  of  inertia  or  inactivity,  considered  in 
the  last  chapter,  are  such  as  may  be  manifested  by  a  single 
insulated  body,  without  reference  to,  or  connection  with,  any 
other  body  whatever.  They  might  all  be  recognised  if  there 
were  but  one  body  existing  in  the  universe.  There  are, 
3* 


30  THE    ELEMENTS    OF    MECHANICS.  CHAP.    IV. 

however,  other  important  results  of  this  law,  to  the  develope- 
ment  of  which  two  bodies  at  least  are  necessary. 

(55.)  If  a  mass  A,  Ji.fr.  4.,  moving  towards  C,  impinge 
upon  an  equal  mass,  which  is  quiescent  at  B,  the  two  masses 
will  move  together  towards  C  after  the  impact,  But  it  will 
be  observed,  that  their  speed  after  the  impact  will  be  only 
half  that  of  A  before  it.  Thus,  after  the  impact,  A  loses 
half  its  velocity  ;  and  B,  which  was  before  quiescent,  re- 
ceives exactly  this  amount  of  motion.  It  appears,  there- 
fore, in  this  case,  that  B  receives  exactly  as  much  motion  as 
A  loses  ;  so  that  the  real  quantity  of  motion  from  B  to  C  is 
the  same  as  the  quantity  of  motion  from  A  to  B. 

Now,  suppose  that  B  consisted  of  two  masses,  each  equal 
to  A,  it  would  be  found  that  in  this  case  the  velocity  of  the 
triple  mass,  after  impact,  would  be  one  third  of  the  velocity 
from  A  to  B.  Thus,  after  impact,  A  loses  two  thirds  of  its 
velocity,  and,  B  consisting  of  two  masses  each  equal  to  A, 
each  of  these  two  receives  one  third  of  A's  motion  ;  so  that 
the  whole  motion  received  by  B  is  two  thirds  of  the  motion 
of  A  before  impact.  By  the  impact,  therefore,  exactly  as 
much  motion  is  received  by  B  as  is  lost  by  A. 

A  similar  result  will  be  obtained,  whatever  proportion  may 
subsist  between  the  masses  A  and  B.  Suppose  B  to  be  ten 
times  A ;  then  the  whole  motion  of  A  must,  after  the  impact, 
be  distributed  among  the  parts  of  the  united  masses  of  A 
and  B  :  but  these  united  masses  are,  in  this  case,  eleven 
times  the  mass  of  A.  Now,  as  they  all  move  with  a  common 
inotjon,  it  follows  that  A's  former  motion  must  be  equally 
distributed  among  them ;  so  that  each  part  shall  have  a.n 
eleventh  part  of  it.  Therefore  the  velocity,  after  impact,  will 
be  the  eleventh  part  of  the  velocity  of  A  before  it.  Thus  A 
loses,  by  the  impact,  ten  eleventh  parts  of  its  motion,  which 
are  precisely  what  B  receives. 

Again,  if  the  masses  of  A  and  B  be  5  and  7,  then  the 
united  mass,  after  impact,  will  be  12.  The  motion  of  A, 
before  impact,  will  be  equally  distributed  between  these  twelve 
parts,  so  that  each  part  will  have  a  twelfth  of  it ;  but  five  of 
these  parts  belong  to  the  mass  A,  and  seven  to  B ;  hence  B 
will  receive  seven  twelfths,  while  A  retains  five  twelfths. 

(56.)  In  general,  therefore,  when  a  mass  A  in  motion 
impinges  on  a  mass  B  at  rest,  to  find  the  motion  of  the  united 
mass  after  impact,  "  divide  the  whole  motion  of  A  into  as 
many  equal  parts  as  there  are  equal  component  masses  in  A 


CHAP.  IV.         ACTION  AND  REACTION.  31 

and  B  together,  and  then  B  will  receive,  by  the  impact,  as 
many  parts  of  this  motion  as  it  has  equal  component  masses." 

This  is  an  immediate  consequence  of  the  property  of  inertia, 
explained  in  the  last  chapter.  If  we  were  to  suppose,  that, 
by  their  mutual  impact,  A  were  to  give  to  B  either  more  or 
less  motion  than  that  which  it  (A)  loses,  it  would  necessarily 
follow,  that  either  A  or  B  must  have  a  power  of  producing  or 
of  resisting  motion,  which  would  be  inconsistent  with  the 
quality  of  inertia  already  defined.  For  if  A  give  to  B  more 
motion  than  it  loses,  all  the  overplus  or  excess  must  be  excited 
in  B  by  the  action  of  A  ;  and,  therefore,  A  is  not  inactive, 
but  is  capable  of  exciting  motion  which  it  does  not  possess. 
On  the  other  hand,  B  cannot  receive  from  A  less  motion  than 
A  loses,  because  then  B  must  be  admitted  to  have  the  power 
by  its  resistance  of  destroying  all  the  deficiency ;  a  power 
essentially  active,  and  inconsistent  with  the  quality  of  inertia. 

(57.)  If  we  contemplate  the  effects  of  impact,  which  we 
have  now  described,  as  facts  ascertained  by  experiment 
(which  they  may  be),  we  may  take  them  as  further  verifica- 
tion of  the  universality  of  the  quality  of  inertia.  But,  on 
the  other  hand,  we  may  view  them  as  phenomena  which  may 
certainly  be  predicted  from  the  -previous  knowledge  of  that 
quality  ;  and  this  is  one  of  .many  instances  of  the  advantage 
which  science  possesses  over  knowledge  merely  practical. 
Having  obtained  by  ^observation  or  experience  a  certain  num- 
ber of  simple  facts,  and  thence  deduced  the  general  qualities 
of  bodies,  we  .are  enabled,  by  demonstrative  reasoning,  to 
discover  other  facts  which  have  never  fallen  under  our  obser- 
vation, or,  if  so,  may  have  never  excited  attention.  In  this 
way,  philosophers  have  discovered  certain  small  motions  and 
slight  changes  which  have  taken  place  among  the  heavenly 
bodies,  and  have  directed  the  attention  of  astronomical  ob- 
servers to  them,  instructing  them  with  the  greatest  precision 
as  to  the  exact  moment  of  time,  and  the  point  of  the  firma- 
ment to  which  they  should  direct  the  telescope,  in  order  to 
witness  the  predicted  event. 

(58.)  Since,  by  the  quality  of  inertia,  a  body  can  neither 
generate  nor  destroy  motion,  it  follows  that  when  two  bodies 
act  upon  each  other  in  any  way  whatever,  the  total  quantity 
of  motion  in  a  given  direction,  after  the  action  takes  place, 
must  be  the  same  as  before  it,  for  otherwise  some  motion 
would  be  produced  by  the  action  of  the  bodies,  which  would 
contradict  the  principle  that  they  are  inert.  The  word  "  ac- 


. 


32  THE  ELEMENTS  OF  MECHANICS.       CHAP.  IV. 

tion"  is  here  applied,  perhaps  improperly,  but  according  to  the 
usuge  of  mechanical  writers,  to  express  a  certain  phenomenon 
or  effect.  It  is,  therefore,  not  to  be  understood  as  imply- 
ing any  active  principle  in  the  bodies  to  which  it  is  attributed. 

(59.)  In  the  cases  of  collision  of  which  we  have  spoken, 
one  of  the  masses  B  was  supposed  to  be  quiescent  before  the 
impact.  We  shall  now  suppose  it  to  be  moving  in  the  same 
direction  as  A,  that  is,  towards  C,  biit  with  a  less  velocity,  so 
that  A  shall  overtake  it,  and  impinge  upon  it.  After  the 
impact,  the  two  masses  will  move  towards  C  wish  a  common 
velocity,  the  amount  of  which  we  now  propose  to  determine. 

If  the  masses  A  and  B  be  equal,  then  their  motions  or 
velocities  added  together  must  be  the  motion  of  the  united 
mass  after  impact,  since  no  motion  can  either  be  created  or 
destroyed  by  that  event.  But  as  A  and  B  move  with  a  com- 
mon motion,  this  sum  must  be  equally  distributed  between 
them,  and  therefore  each  will  move  with  a  velocity  equal  to 
half  the  sum  of  their  velocities  before  the  impact.  Thus,  if  A 
have  the  velocity  7,  and  B  have  5,  the  velocity  of  the  united 
mass,  after  impact,  is  6,  being  the  half  of  12,  the  sum  of  7 
and  5. 

If  A  and  B  be  not  equal,  suppose  them  divided  into  equal 
component  parts,  and  let  A  consist^of  8,  and  B  of  6,  equal 
masses:  let  the  velocity  of  A  be  17,  so  that,  the  motion  of 
each  of  the  8  parts  being  17,  the  motion  of  the  whole  will 
be  136.  In  the  same  manner,  let  the  velocity  of  B  be  10, 
the  motion  of  each  part  being  10,  the  whole  motion  of  the  6 
parts  will  be  60.  The  sum  of  the  two  motions,  therefore, 
towards  C  is  196;  and  since  none  of  this  can  be  lost  by  the 
impact,  nor  any  motion  added  to  it,  this  must  also  be  the 
whole  motion  of  the  united  masses  after  impact.  Being 
equally  distributed  among  the  14  component  parts  of  which 
these  united  masses  consist,  each  part  will  have  a  fourteenth 
of  the  whole  motion.  Hence,  196  being  divided  by  14,  we 
obtain  the  quotient  14,  which  is  the  velocity  with  which  the 
whole  moves.  — £* 

(60.)  In  general,  therefore,  when  two  masses,  moving  in 
the  same  direction,  impinge  one  upon  the  other,  and,  after  im- 
pact, move  together,  their  common  velocity  may  be  determin- 
ed by  the  following  rule  :  "  Express  the  masses  and  velocities 
by  numbers  in  the  usual  way,  and  multiply  the  numbers  ex- 
pressing the  masses  by  the  numbers  which  express  the  veloci- 
ties ;  the  two  products  thus  obtained  being  added  together, 


CHAP.   IV. 


ACTION  AND  REACTION. 


33 


and  their  sum  divided  by  the  sum  of  the  numbers  expressing 
the  masses,  the  quotient  will  be  the  number  expressing  the 
required  velocity." 

(61.)  From  the  preceding  details,  it  appears  that  motion  is 
not  adequately  estimated  by  speed  or  velocity.  For  example, 
a  certain  mass  A,  moving  at  a  determinate  rate,  has  a  certain 
quantity  of  motion.  If  another  equal  mass  B  be  added  to  A, 
and  a  similar  velocity  be  given  to  it,  as  much  more  motion 
will  evidently  be  called  into  existence.  In  other  words,  the  two 
equal  masses  A  and  B  united  have  twice  as  much  motion  as 
the  single  mass  A  had  when  moving  alone,  and  with  the 
same  speed.  The  same  reasoning  will  show  that  three,  equal 
masses  will,  with  the  same  speed,  have  three  times  the  motion 
of  anyone  of  them.  In  general,  therefore,  the  velocity  being 
the  same,  the  quantity  of  motion  will  always  be  increased 
or  diminished  in  the  same  proportion  as  the  mass  moved  is 
increased  or  diminished. 

(02.)  On  the  other  hand,  the  quantity  of  motion  does  not 
depend  on  the  mass  only,  but  also  on  the  speed.  If  a  certain 
determinate  mass  move  with  a  certain  determinate  speed, 
another  equal  mass  which  moves  with  twice  the  speed,  that 
is,  which  moves  over  twice  the  space  in  the  same  time,  will 
have  twice  the  quantity  of  motion.  In  this  manner,  the 
mass  being  the  same,  the  quantity  of  motion  will  increase  or 
diminish  in  the  same  proportion  as  the  velocity. 

(63.)  The  true  estimate,  then,  of  the  quantity  of  motion 
is  found  by  multiplying  together  the  numbers  which  express 
the  mass  and  the  velocity.  Thus,  in  the  example  which  has 
been  last  give*n  of  the  impact  of  masses,  the  quantities  of 
motion  before  and  after  impact  appear  to  be  as  follow  : 


Before  Impact.                     ! 

After  Impact. 
Mass  of  A                  8 

Velocity  of  A  17 

Common  velocity    14 

Quantity  of     )  g       17*  Qr  13(J 
motion  ot  A  ^ 

Quantity  of    ?  fl       ,, 
motion  of  AS     X1 

or    112 

Mass  of  B          G 

Mass  of  B  6 

Velocity  of  B  10 

Common  velocity    14 

Quantity  of    *  6  x  10    or    fj0 
motion  of  B  $ 

Quantity  of    ?        fi 
motion  of  B$ 

14  =  84 

*  The  sign  X  when  placed  between  two  numbers  means  that  they  are  to  be 
multiplied  together. 


34  THE  ELEMENTS  OF  MECHANICS.  CHAP.  IV. 

By  this  calculation  it  appears  that  in  the  impact  A  has  lost 
a  quantity  of  motion  expressed  by  24,  and  that  B  has  re- 
ceived exactly  .that  amount.  The  effect,  therefore,  of  the 
impact  is  a  transfer  of  motion  from  A  to  B  ;  but  no  new 
motion  is  produced  in  the  direction  A  C  which  did  not  exist 
before.  This  is  obviously  consistent  with  the  property  of 
inertiar  and,  indeed,  an  inevitable  result  of  it. 

(64.)  This  phenomenon  is  an  example  of  a  law  deduced 
from  the  property  of  inertia,  and  generally  expressed  thus — 
"  Action  and  reaction  are  equal,  arid  in  contrary  directions." 
The  student  must,  however,  be  cautious  not  to  receive  these 
terms  in  their  ordinary  acceptation.  After  the  full  explana- 
tion of  inertia  given  in  the  last  chapter,  it  is,  perhaps,  scarcely 
necessary  here  to  repeat,  that  in  the  phenomena  manifested 
by  the  motion  of  two  bodies,  there  can  be  neither  "  action" 
nor  "  reaction,"  properly  so  called.  The  bodies  are  absolute- 
ly incapable  either  of  action  or  resistance.  The  ^ense  in 
which  these  words  must  be  received,  as  used  in  the  law,  is 
merely  an  expression  of  the  transfer  of  a  certain  quantity 
of  motion  from  one  body  to  another,  which  is  called  an  action 
in  the  body  which  loses  the  motion,  and  a  reaction  in  the 
body  which  receives  it.  The  accession  of  motion  to  the  latter 
is  said  to  proceed  from  the  action  of  the  former  ;  and  the 
loss  of  the  same  motion  in  the  former  is  ascribed  to  the 
reaction  of  the  latter.  The  whole  phraseology  is,  however, 
most  objectionable  and  unphilosophical,  and  is  calculated  to 
create  wrong  notions. 

(G5.)  The  bodies  impinging  were,  in  the  last  case,  suppos- 
ed to  move  in  the  same  direction.  We  shall  now  consider 
the  case  in  which  they  move  in  opposite  directions. 

First,  let  the  masses  A  and  B  be  supposed  to  be  equal,  and 
moving  in  opposite  directions,  with  the  same  velocity.  Let 
C,  Jig.  5.,  be  the  point  at  which  they  meet.  The  equal 
motions  in  opposite  directions  will,  in  this  case,  destroy  each 
other,  and  both  masses  will  be  reduced  to  a  state  of  rest. 
Thus  the  mass  A  loses  all  its  motion  in  the  direction  A  C, 
which  it  may  be  supposed  to  transfer  to  B  at  the  moment 
of  impact.  But  B,  having  previously  had  an  equal  quantity 
of  motion  in  the  direction  B  C,  will  now  have  two  equal 
motions  impressed  upon  it,  in  directions  immediately  oppo- 
site ;  and,  these  motions  neutralizing  each  other,  the  mass 
becomes  quiescent.  In  this  case,  therefore,  as  in  all  the 
former  examples,  each  body  transfers  to  the  other  all  the 


CHAP.  IV.  ACTION  AND  REACTION.  35 

motion  which  it  loses,  consistently  with  the  principle  of  "  ac- 
tion and  reaction." 

The  masses  A  and  B  being  still  supposed  equal,  let  them 
move  towards  C  with  different  velocities.  Let  A  move  with 
the  velocity  10,  and  B  with  the  velocity  6.  Of  the  10  parts 
of  motion  with  which  A  is  endued,  6  being  transferred  to  B, 
will  destroy  the  equal  velocity  6,  which  B  has  in  the  direction 
B  C.  The  bodies  will  then  move  together  in  the  direction 
C  B,  the  four  remaining  parts  of  A's  motion  being  equally 
distributed  between  them.  Each  body  will,  therefore,  have 
two  parts  of  A's  original  motion,  and  2,  therefore,  will  be  their 
common  velocity  after  impact.  In  this  case,  A  loses  8  of  the 
10  parts  of  its  motion  in  the  direction  A  C.  On  the  other 
hand,  B  loses  the  entire  of  its  6  parts  of  motion  in  the  direc- 
tion B  C,  and  receives  2  parts  in  the  direction  A  C.  This  is 
equivalent  to  receiving  8  parts  of  A's  motion  in  the  direction 
A  C.  Thus,  according  to  the  law  of  "  action  and  reaction," 
B  receives  exactly  what  A  loses.  •.» 

Finally,  suppose  that  both  the  masses  and  velocities  of  A 
and  B  are  unequal.  Let  the  mass  of  A  be  8,  and  its  velocity 
9 ;  and  let  the  mass  of  B  be  6,  and  its  velocity  5.  The 
quantity  of  motion  of  A  will  be  72,  and  that  of  B,  in  the  oppo- 
site direction,  will  be  30.  Of  the  72  parts  of  motion,  which 
A  has  in  the  direction  A  C,  30,  being  transferred  to  B,  will 
destroy  all  its  30  parts  of  motion  in  the  direction  B  C,  and 
the  two  masses  will  move  in  the  direction  C  B,  with  the 
remaining  42  parts  of  motion,  which  will  be  equally  distrib- 
uted among  their  14  component  masses.  Each  component 
part  will,  therefore,  receive  three  parts  of  motion  ;  and  ac- 
cordingly 3  will  be  the  common  velocity  of  the  united  mass 
after  impact. 

(66.)  When  two  masses,  moving  in  opposite  directions, 
impinge  and  move  together,  their  common  velocity  after  im- 
pact may  be  found  by  the  following  rule  : — "  Multiply  the 
numbers  expressing  the  masses  by  those  which  express  the 
velocities  respectively,  and  subtract  the  lesser  product  from 
the  greater  ;  divide  the  remainder  by  the  sum  of  the  num- 
bers expressing  the  masses,  and  the  quotient  will  be  the  com- 
mon velocity  ;  the  direction  will  be  that  of  the  mass  which 
has  the  greater  quantity  of  motion." 

It  may  be  shown  without  difficulty,  that  the  example 
which  we  have  just  given  obeys  the  law  of  "action  and 
reaction." 


THE  ELEMENTS  OF  MECHANICS.  CHAP.  IV. 


Before  Impact. 

Mass  of  A 8 

Velocity  of  A    ...  .  .  9 


Quantity  of  motion  ?  ft  v  nm.7o 


in  direction  A  C 

Mass  of  B .  6 

Velocity  of  B 5 

Quantity  of  motion 

in  direction  B  C 


After  Impact. 

Mass  of  A    8 

Common  velocity  ...  3 


Quantity  of  motion  ?  Q  ^  Q  n  o/i 


hi  direction  A  C 
Mass  of  B    .......  6 

Common  velocity  ...  3 


Quantity  of  motion  ?  - ..  Q  _„  no 
„  .1:     "  *: A  r«      f  °  X  o  Or  18 


hi  direction  A  C 


Hence  it  appears  that  the  quantity  of  motion  in  the  direction 
A  C,  of  which  A  has  been  deprived  by  the  impact,  is  48,  the 
difference  between  72  and  24.  On  the  other  hand,  B  loses 
by  the  impact  the  quantity  30  in  the  direction  B  C,  which  is 
equivalent  to  receiving  30  in  thG  direction  A  C.  But  it  also 
acquires  a  quantity  18  in  the  direction  A  C,  which,  added 
to  the  former  30,  gives  a  total  of  48  received  by  B  in  the 
direction  A  C.  Thus  the  same  quantity  of  motion  which  A 
loses  in  the  direction  A  C,  is  received  by  B  in  the  same 
direction.  The  law  of  "  action  and  reaction"  is,  therefore, 
fulfilled. 

(67.)  The  examples  of  the  equality  of  action  and  reaction 
in  the  collision  of  bodies  may  be  exhibited  experimentally  by 
a  very  simple  apparatus.  Let  A,  Jig.  6.,  and  B  be  two  balls 
of  soft  clay,  or  any  other  substance  which  is  inelastic,  or 
nearly  so,  and  let  these  be  suspended  from  C  by  equal  strings, 
so  that  they  may  be  in  contact ;  and  let  a  graduated  arch, 
of  which  the  centre  is  C,  be  placed  so  that  the  balls  may 
oscillate  over  it.  One  of  the  bulls  being  moved  from  its  place 
of  rest  along  the  arch,  and  allowed  to  descend  upon  the 
other  through  a  certain  number  of  degrees,  will  strike  the 
other  with  a  velocity  corresponding  to  that  number  of  de- 
grees, and  both  balls  will  then  move  together  with  a  velo- 
city which  may  be  estimated  by  the  number  of  degrees  of 
the  arch  through  which  they  rise. 

(68.)  In  all  these  cases  in  which  we  have  explained  the 
law  of  "  action  and  reaction,"  the  transfer  of  motion  from 
one  body  to  the  other  has  been  made  by  impact  or  collision. 
This  phenomenon  has  been  selected  only  because  it  is  the 
most  ordinary  way  in  which  bodies  are  seen  to  affect  each 
other.  The  law  is,  however,  universal,  and  will  be  fulfilled 
in  whatever  manner  the  bodies  may  effect  each  other.  Thus 
A  may  be  connected  with  B  by  a  flexible  string,  which,  at 


CHAP.  IV.  ACTION  AND  REACTION.  37 

the  commencement  of  A's  motion,  is  slack.  Until  the 
string  becomes  stretched,  that  is,  until  A's  distance  from  B 
becomes  equal  to  the  length  of  the  string,  A  will  continue 
to  have  all  the  motion  first  impressed  upon  it.  But  when  the 
string  is  stretched,  a  part  of  that  motion  is  transferred  to  B, 
which  is  then  drawn  after  A  ;  and  whatever  motion  B  in 
this  way  receives,  A  must  lose.  All  that  has  been  observed 
of  the  effect  of  motion  transferred  by  impact  will  be  equally 
applicable  in  this  case. 

Again,  if  B,  jig.  4.,  be  a  magnet  moving  in  the  direction 
B  C  with  a  certain  quantity  of  motion,  and,  while  it  is  so 
moving,  a  mass  of  iron  be  placed  at  rest  at  A,  the  attraction 
of  the  magnet  will  draw  the  iron  after  it  towards  C,  and 
will  thus  communicate  to  the  iron  a  certain  quantity  of  mo- 
tion in  the  direction  of  C.  All  the  motion  thus  communi- 
cated to  the  iron  A  must  be  lost  by  the  magnet  B. 

If  the  magnet  and  the  iron  were  both  placed  quiescent 
at  B  and  A,  the  attraction  of  the  magnet  would  cause  the 
iron  to  move  from  A  towards  B ;  but  the  magnet,  in  this  case, 
not  having  any  motion,  cannot  be  literally  said  to  transfer  a 
motion  to  the  iron.  At  the  moment,  however,  when  the 
iron  begins  to  move  from  A  towards  B,  the  magnet  will  be 
observed  to  begin  also  to  move  from  B  towards  A  ;  and 
if  the  velocities  of  the  two  bodies  be  expressed  by  numbers, 
and  respectively  multiplied  by  the  numbers  expressing  their 
masses,  the  quantities  of  motion  thus  obtained  will  be  found 
to  be  exactly  equal.  We  have  already  explained  why  a  quan- 
tity of  motion  received  in  the  direction  B  A  is  equivalent 
to  the  same  quantity  lost  in  the  direction  A  B.  Hence  it 
appears,  that  the  magnet,  in  receiving  as  much  motion  in 
the  direction  B  A,  as  it  gives  in  the  direction  A  B,  suffers 
an  effect  which  is  equivalent  to  losing  as  much  motion  direct- 
ed towards  C  as  it  has  communicated  to  the  iron  in  the 
same  direction. 

In  the  same  manner,  if  the  body  B  had  any  property  in 
virtue  of  which  it  might  repel  A,  it  would  itself  be  repelled 
with  the  same  quantity  of  motion.  In  a  word,  whatever  be 
the  manner  in  which  the  bodies  may  affect  each  other,  wheth- 
er by  collision,  traction,  attraction,  or  repulsion,  or  by  what- 
ever other  name  the  phenomenon  may  be  designated,  still 
it  is  an  inevitable  consequence,  that  any  motion,  in  a  given 
direction,  which  one  of  the  bodies  may  receive,  must  be 
accompanied  by  a  loss  of  motion  in  the  same  direction,  and 

4    . 


38  THE  ELEMENTS  OF  MECHANICS.  CHAP.  ty*. 

-      *'         '  f 

to  tne  same  amount,  by  the  other  body,  or  the  acquisition 
of  as  much  motion  in  the  contrary  direction  ;  or,  finally,  by 
a  loss  in  the  same  direction,  and  an  acquisition  of  motion  in 
the  contrary  direction,  the  combined  amount  of  which  is 
equal  to  the  motion  received  by  the  former. 

(69.)  From  the  principle,  that  the  force  of  a  body  in  mo- 
tion depends  on  the  mass  and  the  velocity,  it  follows,  that 
any  body,  however  small,  may  be  made  to  move  with  the 
same  force  as  any  other  body,  however  great,  by  giving  to  the 
smaller  body  a  velocity  which  bears  to  that  of  the  greater 
the  same  proportion  as  the  mass  of  the  greater  bears  to  the 
mass  of  the  smaller.  Thus  a  feather,  ten  thousand  of  which 
would  have  the  same  weight  as  a  cannon-ball,  would  move 
with  the  same  force  if  it  had  ten  thousand  times  the  velocity ; 
and,  in  such  a  case,  these  two  bodies,  encountering  in  oppo- 
site directions,  would  mutually  destroy  each  other's  mo- 
tion. 

(70.)  The  consequences  of  the  property  of  inertia,  which 
have  been  explained  in  the  present  and  preceding  chapters, 
have  been  given  by  Newton,  in  his  PRINCIPIA,  and,  after  him, 
in  most  English  treatises  on  mechanics,  under  the  form  of 
three  propositions,  which  are  called  the  "  laws  of  motion." 
They  are  as  follow  : — 

I. 

"  Every  body  must  persevere  in  its  state  of  rest,  or  of 
uniform  motion  in  a  straight  line,  unless  it  be  compelled 
to  change  that  state  by  forces  impressed  upon  it." 

It 

"  Every  change  of  motion  must  be  proportional  to  the 
impressed  force,  and  must  be  in  the  direction  of  that  straight 
line  in  which  the  force  is  impressed." 

III. 

"  Action  must  always  be  equal,  and  contrary  to  reaction ; 
or  the  actions  of  two  bodies  upon  each  other  must  be  equal, 
and  directed  towards  contrary  sides." 

When  inertia  and  force  are  defined,  the  first  law  becomes 
an  identical  proposition.  The  second  law  cannot  be  render- 
ed perfectly  intelligible  until  the  student  has  read  the  chapter 
on  the  composition  and  resolution  of  forces,  for,  in  fact,  it  is 
intended  as  an  expression  of  the  whole  body  of  results  in 


CHAP.  IV.  FAMILIAR  ILLUSTRATIONS.  39 

that  chapter.  The  third  law  has  been  explained  in  the 
present  chapter  as  far  as  it  can  be  rendered  intelligible  in 
the  present  stage  of  our  progress. 

We  have  noticed  these  formularies  more  from  a  respect 
for  the  authorities  by  which  they  have  been  adopted,  than 
from  any  persuasion  of  their  utility.  Their  full  import  can- 
not be  comprehended  until  nearly  the  whole  of  elementary 
mechanics  has  been  acquired,  and  then  all  such  summaries 
become  useless. 

(71.)  The  consequences  deduced  from  the  consideration 
of  the  quality  of  inertia  in  this  chapter,  will  account  for  many 
effects  which  fall  under  our  notice  daily,  and  with  which  we 
have  become  so  familiar,  that  they  have  almost  ceased  to 
excite  curiosity.  One  of  the  facts  of  which  we  have  most 
frequent  practical  illustration  is,  that  the  quantity  of  motion, 
or  moving  fore*,  as  it  is  sometimes  called,  is  estimated  by 
the  velocity  of  the  motion,  and  the  weight  or  mass  of  the 
thing  moved,  conjointly. 

If  the  same  force  impel  two  balls,  one  of  one  pound  weight, 
and  the  other  of  two  pounds,  it  follows,  since  the  balls  can 
neither  give  force  to  themselves  nor  resist  that  which  is  im- 
pressed upon  them,  that  they  will  move  with  the  same  force. 
But  the  lighter  ball  will  move  with  twice  the  speed  of  the 
heavier.  The  impressed  force  which  is  manifested  by  giving 
velocity  to  a  double  mass  in  the  one,  is  engaged  in  giving  a 
double  velocity  to  the  other. 

If  a  cannon-ball  were  forty  times  the  weight  of  a  musket- 
ball,  but  the  musket-ball  moved  with  forty  times  the  velocity 
of  the  cannon-ball,  both  would  strike  any  obstacle  with  the 
same  force,  and  would  overcome  the  same  resistance  ;  for 
the  one  would  acquire  from  its  velocity  as  much  force  as  the 
other  derives  from  its  weight. 

A  very  small  velocity  may  be  accompanied  by  enormous 
force,  if  the  mass  which  is  moved  with  that  velocity  be  propor- 
tionally great.  A  large  ship  floating  near  the  pier  wall,  may 
approach  it  with  so  small  a  velocity  as  to  be  scarcely  per- 
ceptible, and  yet  the  force  will  be  so  great  as  to  crush  a 
small  boat. 

A  grain  of  shot  flung  from  the  hand,  and  striking  the 
person,  will  occasion  no  pain,  and,  indeed,  will  scarcely  be 
nit,  while  a  block  of  stone  having  the  same  velocity  would 
occasion  death. 


40 


THE   ELEMENTS  OF  MECHANICS.  CHAP.   IV 


If  a  body  in  motion  strike  a  body  at  rest,  the  striking 
body  must  sustain  as  great  a  shock  from  the  collision  as  if  it 
had  been  at  rest,  and  struck  by  the  other  body  with  the  same 
force.  For  the  loss  of  force  which  it  sustains  in  the  one 
direction,  is  an  effect  of  the  same  kind  as  if,  being  at  rest, 
it  had  received  as  much  force  in  the  opposite  direction.  If  a 
man,  walking  rapidly,  or  running,  encounters  another  stand- 
ing still,  he  suffers  as  much  from  the  collision  as  the  man 
against  whom  he  strikes. 

If  a  leaden  bullet  be  discharged  against  a  plank  of  hard 
wood,  it  will  be  found  that  the  round  shape  of  the  ball  is 
destroyed,  and  that  it  has  itself  suffered  a  force  by  the  im- 
pact, which  is  equivalent  to  the  effect  which  it  produces  upon 
the  plank. 

When  two  bodies  moving  in  opposite  directions  meet,  each 
body  sustains  as  great  a  shock  as  if,  being  at  rest,  it  had  been 
struck  by  the  other  body  with  the  united  forces  of  the  two. 
Thus,  if  two  equal  balls,  moving  at  the  rate  of  ten  feet  in 
a  second,  meet,  each  will  be  struck  with  the  same  force  as 
if,  being  at  rest,  the  other  had  moved  against  it  at  the  rate 
of  twenty  feet  in  a  second.  In  this  case,  one  part  of  the 
shock  sustained  arises  from  the  loss  of  force  in  one  direction, 
and  another  from  the  reception  of  force  in  the  opposite 
direction. 

For  this  reason,  two  persons  walking  in  opposite  directions 
receive  from  their  encounter  a  more  violent  shock  than  might 
be  expected.  If  they  be  of  nearly  equal  weight,  and  one  be 
walking  at  the  rate  of  three  and  the  other  four  Nmiles  an 
hour,  each  sustains  the  same  shock  as  if  he  had  been  at  rest, 
and  struck  by  the  other  running  at  the  rate  of  seven  miles  an 
hour. 

This  principle  accounts  for  the  destructive  effects  arising 
from  ships  running  foul  of  each  other  at  sea.  If  two  ships 
of  500  tons  burden  encounter  each  other,  sailing  at  ten  knot*? 
an  hour,  each  sustains  the  shock  which,  being  at  rest,  it 
would  receive  from  a  vessel  of  1000  tons  burden  sailing  ten 
knots  an  hour. 

It  is  a  mistake  to  suppose,  that  when  a  large  and  small 
body  encounter,  the  small  body  suffers  a  greater  shock  than 
the  large  one.  The  shock  which  they  sustain  must  be  the 
same  ;  but  the  large  body  may  be  better  able  to  bear  it. 

When  the  fist  of  a  pugilist  strikes  the  body  of  his  an- 
tagonist, it  sustains  as  great  a  shock  as  it  gives  ;  but  the 


CHAP.  V.   COMPOSITION  AND  RESOLUTION  OF  FORCE.        41 

part  being  more  fitted  to  endure  the  blow,  the  injury  and 
pain  are  inflicted  on  his  opponent.  This  is  not  the  case, 
however,  when  fist  meets  fist.  Then  the  parts  in  collision 
are  equally  sensitive  and  vulnerable,  and  the  effect  is  aggra- 
vated by  both  having  approached  each  other  with  great  force. 
The  effect  of  the  blow  is  the  same  as  if  one  fist,  being  held 
at  rest,  were  struck  by  the  other  with  the  combined  force  of 
both. 


CHAPTER  V. 

THE  COMPOSITION  AND  RESOLUTION   OF  FORCE. 

(72.)  MOTION  and  pressure  are  terms  too  familiar  to  need 
explanation.  It  may  be  observed,  generally,  that  definitions 
in  the  first  rudiments  of  a  science  are  seldom,  if  ever,  com- 
prehended. The  force  of  words  is  learned  by  their  applica- 
tion ;  and  it  is  not  until  a  definition  becomes  useless,  that 
we  are  taught  the  meaning  of  the  terms  in  which  it  is  ex- 
pressed. Moreover,  we  are  perhaps  justified  in  saying,  that, 
in  the  mathematical  sciences,  the  fundamental  notions  are 
of  so  uncompounded  a  character,  that  definitions,  when  de- 
veloped and  enlarged  upon,  often  draw  us  into  metaphysical 
subtleties  and  distinctions,  which,  whatever  be  their  merit  or 
importance,  would  be  here  altogether  misplaced.  We  shall, 
therefore,  at  once  take  it  for  granted,  that  the  words  motion 
and  pressure  express  phenomena  or  effects  which  are  the 
subjects  of  constant  experience  and  hourly  observation  ;  and 
if  the  scientific  use  of  these  words  be  more  precise  than 
their  general  and  popular  application,  that  precision  will 
soon  be  learned  by  their  frequent  use  in  the  present  treatise. 

(73.)  FORCE  is  the  name  given  in  mechanics  to  whatever 
produces  motion  or  pressure.  This  word  is  also  often  used 
to  express  the  motion  or  pressure  itself;  and  when  the  cause 
of  the  motion  or  pressure  is  not  known,  this  is  the  only  cor- 
rect use  of  the  word.  Thus,  when  a  piece  of  iron  moves 
toward  a  magnet,  it  is  usual  to  say  that  the  cause  of  the  motion 
is  "the  attraction  of  the  magnet;"  but  in  effect  we  are  igno- 
rant of  the  muse  of  this  phenomenon  ;  and  the  name  attrac- 
tion would  be  better  applied  to  the  effect,  of  which  we  have 
exnerience.  In  like  manner  the  attraction  and  repulsion  of 
4* 


4*2  THE    ELEMENTS    OF    MECHANICS.  CHAP.    V 

electrified  bodies  should  be  understood,  not  as  names  for  un- 
known causes,  but  as  words  expressing  observed  appearances 
or  effects. 

When  a  certain  phraseology  has,  however,  gotten  into  gen- 
eral use,  it  is  neither  easy  nor  convenient  to  supersede  it. 
We  shall,  therefore,  be  compelled,  in  speaking  of  motion  and 
pressure,  to  use  the  language  of  causation ;  but  must  advise 
the  student  that  it  is  effects,  and  not  causes,  which  will  be 
expressed. 

(74.)  If  two  forces  act  upon  the  same  point  of  a  body  in 
different  directions,  a  single  force  may  be  assigned,  which, 
acting  on  that  point,  will  produce  the  same  result  as  the 
united  effects  of  the  other  two. 

Let  P,Jig-  7.,  be  the  point  on  which  the  two  forces  act, 
and  let  their  directions  be  P  A  and  P  B.  From  the  point  P, 
upon  the  line  P  A,  take  a  length  P  «,  consisting  of  as  many 
inches  as  there  are  ounces  in  the  force  P  A  :  and,  in  like 
manner,  take  P  £>,  in  the  direction  P  B,  consisting  of  as  many 
inches  as  there  are  ounces  in  the  force  P  B.  Through  a 
draw  a  line  parallel  to  P  B,  and  through  b  draw  a  line  parallel 
to  P  A,  and  suppose  that  these  lines  meet  at  c.  Then  draw 
PC.  A  single  force,  acting  in  the  direction  P  C,  and  con- 
sisting of  as  many  ounces  as  the  line  P  c.  consists  of  inches, 
will  produce  upon  the  point  P  the  same  effect  as  the  two 
forces  P  A  and  P  B  produce  acting  together. 

(75.)  The  figure  P  a  c  b  is  called,  in  GEOMETRY,  a  parallel- 
ogram ;  the  lines  P  «,  P  b,  are  called  its  sides,  and  the  line  P 
c  is  called  its  diagonal.  Thus  the  method  of  finding  an 
equivalent  for  two  forces,  which  we  have  just  explained,  is 
generally  called  "  the  parallelogram  of  forces,"  and  is  usually 
expressed  thus  :  "  If  two  forces  be  represented  in  quantity 
and  direction  by  the  sides  of  a  parallelogram,  an  equivalent 
force  will  be  represented  in  quantity  and  direction  by  its 
diagonal." 

(70.)  A  single  force,  which  is  thus  mechanically  equivalent 
to  two  or  more  other  forces,  is  called  their  resultant,  and 
relatively  to  it  they  are  called  its  components.  In  any  me- 
chanical investigation,  when  the  result  is  used  for  the  compo- 
nents, which  it  always  may  be,  the  process  is  called  "  the 
composition  of  force."  It  is,  however,  frequently  expedient 
to  substitute  for  a  single  force  two  or  more  forces,  to  which 
it  is  mechanically  equivalent,  or  of  which  it  is  the  resultant. 
This  process  is  called  •'  the  resolution  of  force." 


CHAP.  V.         PARALLELOGRAM  OF  FORCES.  43 

(77.)  To  verify  experimentally  the  theorem  of  the  parallel- 
ogram of  forces  is  not  difficult.  Let  two  small  wheels,  M  N, 
Jig.  8,  with  grooves  in  their  edges  to  receive  a  thread,  be 
attached  to  an  upright  board,  or  to  a  wall.  Let  a  thread  be 
passed  over  them,  having  weights,  A  and  B,  hooked  upon 
loops  at  its  extremities.  From  any  part  P  of  the  thread  be- 
tween the  wheels  let  a  weight.  C  be  suspended :  it  will  draw 
the  thread  downwards,  so  as  to  form  an  angle  M  P  N,  and 
the  apparatus  will  settle  itself  at  rest  in  some  determinate  po- 
sition. In  this  state  it  is  evident  that,  since  the  weight  C, 
acting  in  the  direction  P  C,  balances  the  weights  A  and  B, 
acting  in  the  directions  P  M  and  P  N,  these  two  forces  must 
be  mechanically  equivalent  to  a  force  equal  to  the  weight  C, 
and  acting  directly  upwards  from  P.  The  weight  C  is  there- 
fore the  quantity  of  the  resultant  of  the  forces  P  M  and  P  N; 
and  the  direction  of  the  resultant  is  that  of  a  line  drawn 
directly  upwards  from  P. 

To  ascertain  how  far  this  is  consistent  with  the  theorem  of 
"  the  parallelogram  of  forces,"  let  a  line  P  O  be  drawn  upon 
the  upright  board  to  which  the  wheels  are  attached,  from  the 
point  P  upward,  in  the  direction  of  the  thread  C  P.  Also, 
let  lines  be  drawn  upon  the  board  immediately  under  the 
threads  P  M  and  P  N.  From  the  point  P,  on  the  line  P  O, 
take  as  many  inches  as  there  are  ounces  in  the  weight  C. 
Let  the  part  of  P  O  thus  measured  be  P  c,  and  from  c  draw 
c  a  parallel  to  P  N,  and  c  5  parallel  to  P  M.  If  the  sides 
P  «  and  P  b  of  the  parallelogram,  thus  formed,  be  measured, 
it  will  be  found  that  P  a  will  consist  of  as  many  inches  as 
here  are  ounces  in  the  weight  A',  and  P  b  of  as  many  inches 
s  there  are  ounces  in  the  weight  B. 

In  this  illustration,  ounces  and  inches  have  been  used  as  the 
subdivisions  of  ibcigkt  and  length.  It  is  scarcely  necessary  to 
state,  that  any  other  measures  of  these  quantities  would  serve 
as  well,  only  observing  that  the  same  denominations  must 
be  preserved  in  all  parts  of  the  same  investigation. 

(78.)  Among  the  philosophical  apparatus  of  the  University 
of  London,  is  a  very  simple  and  convenient  instrument  which 
I  have  constructed  for  the  experimental  illustration  of  this  im- 
portant theorem.  The  wheels  M  N  are  attached  to  the  tops 
of  two  tall  stands,  the  heights  of  which  may  be  varied  at 
pleasure  by  an  adjusting  screw.  A  jointed  parallelogram, 
A  B  C  D,  fig.  9.,  is  formed,  whose  sides  are  divided  into 


44  THE  ELEMENTS  OF  MECHANICS.       CHAP.  V 

inches,  and  the  joints  at  A  and  B  are  movable,  so  as  to  vary 
the  lengths  of  the  sides  at  pleasure.  The  joint  C  is  fixed  at 
the  extremity  of  a  ruler,  also  divided  into  inches,  while  the 
opposite  joint  A  is  attached  to  a  brass  loop,  which  surrounds 
the  diagonal  ruler  loosely,  so  as  to  slide  freely  along  it.  An 
adjusting  screw  is  provided  in  this  loop  so  as  to  clamp  it  in 
any  required  position. 

In  making  the  experiment,  the  sides  A  B  and  A  D,  C  B 
and  C  D,  are  adjusted  by  the  joints  B  and  A  to  the  same 
number  of  inches  respectively  as  there  are  ounces  in  the 
weights  A  and  B,^'.  8.  Then  the  diagonal  A  C  is  adjusted 
by  the  loop  and  screw  at  A,  to  as  m;my  inches  as  there  are 
ounces  in  the  weight  C.  This  done,  the  point  A  is  placed 
behind  ¥,jig.  8.,  and  the  parallelogram  is  held  upright,  so 
that  the  diagonal  A  C  shall  be  in  the  direction  of  the  vertical 
thread  P  C.  The  sides  A  B  and  A  D  will  then  be  found  to 
take  the  direction  of  the  threads  P  M  and  P  N.  By  chang- 
ing the  weights  and  the  lengths  of  the  diagonal  and  sides  of 
the  parallelogram,  the  experiment  may  be  easily  varied  at 
pleasure. 

(79.)  In  the  examples  of  the  composition  of  forces  which 
we  have  here  given,  the  effects  of  the  forces  are  the  produc- 
tion of  pressures ;  or,  to  speak  more  correctly,  the  theorem 
which  we  have  illustrated,  is  "  the  composition  of  pressures." 
For  the  point  P  is  supposed  to  be- at  rest,  and  to  be  drawn  or 
pressed  in  the  directions  P  M  and  P  N.  In  the  definition 
which  has  been  given, of  the  word  force,  it  is  declared  to 
include  motions  as  well  as  pressures.  In  fact,  if  motion  be 
resisted,  the  effect  is  converted,  into  pressure.  The  same 
cause,  acting  upon  a  body,  will  either  produce  motion  or  pres- 
sure, according  as  the  body  is  free  or  restrained.  If  the 
body  be  free,  motion  ensues;  if  restrained,  pressure,  or  both 
these  effects  together.  It  is  therefore  consistent  with  analo- 
gy to  expect  that  the  same  theorems  which  regulate  pressures, 
will  also  be  applicable  to  motions ;  and  we  find  accordingly 
a  most  exact  correspondence.  ' 

(80.)  If  a  body  have  a  motion  in  the  direction  A  B,  and 
at  the  point  P  it  receive  another  motion,  such  as  wrould  carry 
it  in  the  direction  P  C,Jig.  10.,  were  it  previously  quiescent 
at  P,  it  is  required  to  determine  the  direction  which  the  body 
will  take,  and  the  speed  with  which  it  will  move,  under  these 
circumstances. 


CHAP.    V.  COMPOSITION    OF    FORCES.  45 

Let  the  velocity  with  which  the  body  is  moving  from  A  to 
B  be  such,  that  it  would  move  through  a  certain  space,  sup- 
pose P  N,  in  one  second  of  time,  and  let  the  velocity  of  the 
motion  impressed  upon  it  at  P  be  such,  that,  if  it  had  no 
previous  motion,  it  would  move  from  P  to  M  in  one  second. 
From  the  point  M  draw  a  line  parallel  to  P  B,  and  from  N 
draw  a  line  parallel  to  P  C,  and  suppose  these  lines  to  meet 
at  some  point,  as  O.  Then  draw  the  line  P  0.  In  conse- 
quence of  the  two  motions,  which  are  at  the  same  time 
impressed  upon  the  body  at  P,  it  will  move  in  the  straight 
line  from  P  to  0. 

Thus  the  two  motions,  which  are  expressed  in  quantity  and 
direction  by  the  sides  of  a  parallelogram,  will,  when  given  to 
the  same  body,  produce  a  single  motion,  expressed  in  quanti- 
ty and  direction  by  its  diagonal ;  a  theorem  which  is  to 
motions  exactly  what  the  former  was  to  pressures. 

There  are  various  methods  of  illustrating  experimentally 
the  composition  of  motion.  An  ivory  ball,  being  placed  upon 
a  perfectly  level,  square  table,  at  one  of  the  corners,  and 
receiving  two  equal  impulses,  in  the  directions  of  the  sides 
of  the  table,  will  .move  along  the  diagonal.  Apparatus  for 
this  experiment  differ  from  each  other  only  in  the  way  of 
communicating  the  impulses  to  the  ball. 

(81.)  As  two  motions  simultaneously  communicated  to  a 
body  are  equivalent  to  a  single  motion  in  an  intermediate 
direction,  so  also  a  single  motion  may  be  mechanically  re- 
placed, by  two  motions  in  directions  expressed  by  the  sides  of 
any  parallelogram,  whose  diagonal  represents  the  single  mo- 
tion. This  process  is  "  the  resolution  of  motion,"  and  gives 
considerable  clearness  and  facility/ to  many  mechanical  inves- 
tigations. 

(82.)  It  is  frequently  necessary  to  express  the  portion  of  a 
given  force,  which  acts  in  some  given  direction  different  from 
the  immediate  direction  of  the  force  itself.  Thus,  if  a  force 
act  from  A.,  Jig.  11.,  in  the  direction  A  C,  we  may  require  to 
estimate  what  part  of  thnt  force  acts  in  the  direction  A  B. 
If  the  force  be  a  pressure,  take  as  many  inches  A  P  from  A, 
on  the  line  A  C,  as  there  are  ounces  in  the  force,  and  from  P 
draw  P  M  perpendicular  to  A  B ;  then  the  part,  of  the  force 
which  acts  along  A  B  will  be  as  many  ounces  as  there  are 
inches  in  A  M.  The  force  A  B  is  mechanically  equivalent  to 
two  forces,  expressed  by  the  sides  A  M.and  A  N  of  the  par- 
allelogram :  but  A  N,  boing  perpendicular  to  A  B,  can  have 


46  THE    ELEMENTS    OF    MECHANICS.  CHAP.    V. 

no  effect  on  a  body  at  A,  in  the  direction  of  A  B,  and  there- 
fore the  effective  part  of  the  force  A  P  in  the  direction  A  B 
is  expressed  by  A  M. 

(83.)  Any  number  of  forces  acting  on  the  same  point  of  a 
body  may  be  replaced  by  a  single  force,  which  is  mechanical- 
ly equivalent  to  them,  and  which  is,  therefore,  their  resultant. 
This  composition  may  be  effected  by  the  successive  applica- 
tion of  the  parallelogram  of  forces.  Let  the  several  forces 
be  called  A,  B,  C,  D,  E,  &c.  Draw  the  parallelogram  whose 
sides  express  the  forces  A  and  B,  and  let  its  diagonal  be  A'. 
The  force  expressed  by  A'  will  be  equivalent  to  A  and  B. 
Then  draw  the  parallelogram  whose  sides  express  the  forces 
A'  and  C,  and  let  its  diagonal  be  B7.  This  diagonal  will 
express  a  force  mechanically  equivalent  to  A'  and  C.  But 
A'  is  mechanically  equivalent  to  A  and  B,  and  therefore  B7  is 
mechanically  equivalent  to  A,  B,  and  C.  Next  construct  a 
parallelogram,  whose  sides  express  the  forces  B'  and  D,  and 
let  its  diagonal  be  C'.  The  force  expressed  by  C'  will  be 
mechanically  equivalent  to  the  forces  B'  and  D;  but  the 
force  B'  is  equivalent  to  A,  B,  C,  and  therefore  C'  is  equiva- 
lent to  A,  B,  C,  and  D.  By  continuing  this  process,  it  is 
evident,  that  a  single  force  may  be  found,  which  will  be 
equivalent  to,  and  may  be  always  substituted  for,  any  number 
of  forces  which  act  upon  the  same  point. 

If  the  forces  which  act  upon  the  point  neutralize  each 
other,  so  that  no  motion  can  ensue,  they  are  said  to  be  in 
equilibrium. 

(84.)  Examples  of  the  composition  of  motion  and  pressure 
are  continually  presenting  themselves.  They  occur  in  almost 
every  instance  of  motion  or  force  which  falls  under  our  ob- 
servation. The  difficulty  is,  to  find  an  example  which,  strict- 
"y  speaking,  is  a  simple  motion. 

When  a  boat  is  rowed  across  a  river,  in  which  there  is  a 
current,  it  will  not  move  in  the  direction  in  which  it  is  im- 
pelled by  the  oars.  Neither  will  it  take  the  direction  of  the 
stream,  but  will  proceed  exactly  in  that  intermediate  direction 
which  is  determined  by  the  composition  of  force. 

Let  A, Jig.  12.,  be  the  place  of  the  boat  at  starting;  and 
suppose  that  the  oars  are  so  worked  as  to  impel  the  boat  to- 
wards B  with  a  force  which  would  carry  it  to  B  in  one  hour, 
if  there  were  no  current  in  the  river.  But,  on  the  other  hand, 
suppose  the  rapidity  of  the  current  is  such,  that,  without  any 


CHAP.    V.  FAMILIAR    EXAMPLES.  47 

exertion  of  the  rowers,  the  boat  would  float  down  the  stream 
in  one  hour  to  C.  From  C  draw  C  D  parallel  to  A  B,  and 
draw  the  straight  line  A  D  diagonally.  The  combined  effect 
of  the  oars  and  the  current  will  be,  that  the  boat  will  be  car- 
ried along  A  D,  and  will  arrive  at  the  opposite  bank  in  one 
hour,  at  the  point  D. 

If  the  object  be,  therefore,  to  reach  the  point  B,  starting 
from  A,  the  rowers  must  calculate,  as  nearly  as  possible,  the 
velocity  of  the  current.  They  must  imagine  a  certain  point 
E  at  such  a  distance  above  B  that  the  boat  would  be  floated 
by  the  stream  from  E  to  B  in  the  time  taken  in  crossing  the 
river  in  the  direction  A  E,  if  there  were  no  current.  If  they 
row  towards  the  point  E,  the  boat  will  arrive  at  the  point  B, 
moving  in  the  line  A  B. 

In  this  case,  the  boat  is  impelled  by  two  forces,  that  of  the 
oars  in  the  direction  A  E,  and  that  of  the  current  in  the  di- 
rection A  C.  The  result  will  be,  according  to  the  parallelo- 
gram of  forces,  a  motion  in  the  diagonal  A  B. 

The  wind  and  tide  acting  upon  a  vessel  is  a  case  of  a 
similar  kind.  Suppose  that  the  wind  is  made  to  impel  the 
vessel  in  the  direction  of  the  keel  ;  while  the  tide  may  be 
acting  in  any  direction  oblique  to  that  of  the  keel.  The 
course  of  the  vessel  is  determined  exactly  in  the  same  man- 
ner as  that  of  the  boat  in  the  last  example. 

The  action  of  the  oars  themselves,  in  impelling  the  boat,  is 
an  example  of  the  composition  of  force.  Let  A,  jig.  13.,  be 
the  head,  and  B  the  stern  of  the  boat.  The  boatman  pre- 
sents his  face  towards  B,  and  places  the  oars  so  that  their 
blades  press  against  the  water  in  the  directions  C  E,  D  F. 
The  resistance  of  the  water  produces  forces  on  the  side  of 
the  boat,  in  the  directions  G  L  and  H  L,  which,  by  the  com- 
position of  force,  are  equivalent  to  the  diagonal  force  K  L, 
in  the  direction  of  the  keel. 

Similar  observations  will  apply  to  almost  every  body,  im- 
pelled by  instruments  projecting  from  its  sides,  and  acting 
against  a  fluid.  The  motions  of  fishes,  the  act  of  swimming, 
the  flight  of  birds,  are  all  instances  of  the  same  kind. 

(85.)  The  action  of  wind  upon  the  sails  of  a  vessel,  and 
the  force  thereby  transmitted  to  the  keel,  modified  by  the 
rudder,  is  a  problem  which  is  solved  by  the  principles  of  the 
composition  and  resolution  of  force ;  but  it  is  of  too  compli- 
cated and  difficult  a  nature  to  be  introduced  with  all  its 
necessary  conditions  and  limitations  in  this  place.  The 


48  THE    ELEMENTS    OF    MECHANICS.  CHAP.    V. 

question  may,  however,  be  simplified,  if  we  consider  the 
canvass  of  the  sails  to  be  stretched  so  completely  as  to  form 
a  plane  surface.  Let  A  B,^.  14.,  be  the  position  of  the  sail, 
and  let  the  wind  blow  in  the  direction  C  D.  If  the  line  C  D 
be  taken  to  express  the  force  of  the  wind,  let  D  E  C  F  be  a 
parallelogram,  of  which  it  is  the  diagonal.  The  force  C  J) 
is  equivalent  to  two  forces,  one  in  the  direction  F  D  of  the 
plane  of  the  canvass,  and  the  other  E  D  perpendicular  to  the 
sail.  The  effect,  therefore,  is  the  same  as  if  there  were  two 
winds,  one  blowing  in  the  direction  of  F  D  or  B  A,  that  is, 
against  the  edge  of  the  sail,  and  the  other,  E  D,  blowing  full 
against  its  face.  It  is  evident  that  the  former  will  produce 
no  effect  whatever  upon  the  sail,  and  that  the  latter  will  urge 
the  vessel  in  the  direction  D  G. 

Let  us  now  consider  this  force  D  G  as  acting  in  the  diago- 
nal of  the  parallelogram  D  H  G  I.  It  will  be  equivalent  to 
two  forces,  D  H  and  D  I,  acting  along  the  sides.  One  of 
these  forces,  D  H,  is  in  the  direction  of  the  keel,  and  the 
other,  D  I,  at  right  angles  to  the  length  of  the  vessel,  so  as 
to  urge  it  sideways.  The  form  of  the  vessel  is  evidently 
such  as  to  offer  a  great  resistance  to  the  latter  force,  and  very 
little  to  the  former.  It  consequently  proceeds  with  consider- 
able velocity  in  the  direction  D  H  of  its  keel,  and  makes 
way  very  slowly  in  the  sideward  direction  D  I.  The  latter 
effect  is  called  lee-way. 

From  this  explanation  it  will  be  easily  understood,  how  a 
wind  which  is  nearly  opposed  to  the  course  of  a  vessel  may, 
nevertheless,  be  made  to  impel  it  by  the  effect  of  sails.  The 
angle  B  D  V,  formed  by  the  sail  and  the  direction  of  the 
keel,  may  be  very  oblique,  as  may  also  be  the  angle  C  D  B 
formed  by  the  direction  of  the  wind  and  that  of  the  sail. 
Therefore  the  angle  C  D  V,  made  up  of  these  two,  and  which 
is  that  formed  by  the  direction  of  the  wind  and  that  of  the 
keel,  may  be  very  oblique.  In  Jig-  15.  the  wind  is  nearly 
contrary  to  the  direction  of  the  keel,  and  yet  there  is  an 
impelling  force  expressed  by  the  line  D  H,  the  line  C  D  ex- 
pressing, as  before,  the  whole  force  of  the  wind. 

In  this  example  there  are  two  successive  decompositions 
of  force.  First,  the  original  force  of  the  wind  C  I)  is  re- 
solved into  two,  E  D  and  F  D ;  and  next  the  element  E  D, 
or  its  equal  D  G,  is  resolved  into  D  I  and  D  H  ;  so  that  the 
original  force  is  resolved  into  three,  viz.  F  D,  D  I,  D  H, 
which,  taken  together,  are  mechanically  equivalent  to  it. 


CHAP.    V.  FAMILIAR    EXAMPLES.  49 

The  part  F  D  is  entirely  ineffectual ;  it  glides  off  on  the  sur- 
face of  the  canvass  without  producing  any  effect  upon  the 
vessel.  The  part  D  I  produces  Ice-way,  and  the  part  D  II 
impels. 

(86.)  If  the  wind,  however,  be  directly  contrary  to  the 
course  which  it  is  required  that  the  vessel  should  take,  there 
is  no  position  which  can  be  given  to  the  sails  which  will  im- 
pel the  vessel.  In  this  case,  the  required  course  itself  is 
resolved  into  two,  in  which  the  vessel  sails  alternately,  a 
process  which  is  called  tacking.  Thus,  suppose  the  vessel  is 
required  to  move  from  A  to  E,^>\  16.,  the  wind  setting  from 
E  to  A.  The  motion  A  B  being  resolved  into  two,  by  being 
assumed  as  the  diagonal  of  a  parallelogram,  the  sides  A  a 
a  B  of  the  parallelogram  are  successively  sailed  over,  and 
the  vessel  by  this  means  arrives  at  B,  instead  of  moving  along 
the  diagonal  A  B.  In  the  same  manner  she  moves  along  B  b, 
b  C,  C  c,  c  D,  D  d,  d  E,  and  arrives  at  E.  She  thus  sails 
continually  at  a  sufficient  angle  with  the  wind  to  obtain  an 
impelling  force,  yet  at  a  sufficiently  small  angle  to  make  way 
in  her  proposed  course. 

The  consideration  of  the  effect  of  the  rudder,  which  we 
have  omitted  in  the  preceding  illustration,  affords  another 
instance  of  the  resolution  of  force.  We  shall  not,  however, 
pursue  this  example  further. 

(87.)  A  body  falling  from  the  top  of  the  mast  when  the 
vessel  is  in  full  sail,  is  an  example  of  the  composition  of  mo- 
tion. It  might  be  expected,  that,  during  the  descent  of  the 
body,  the  vessel,  having  sailed  forward,  would  leave  it  behind, 
and  that,  therefore,  it  would  fall  in  the  water  behind  the 
stern,  or  at  least  on  the  deck,  considerably  behind  the  mast. 
On  the  other  hand,  it  is  found  to  fall  at  the  foot  of  the  mast, 
exactly  as  it  would  if  the  vessel  were  not  in  motion.  To 
account  for  this,  let  A  B,  jig.  17.,  be  the  position  of  the 
mast  when  the  body  at  the  top  is  disengaged.  The  mast  is 
moving  onwards  with  the  vessel  in  the  direction  A  C,  so  that 
in  the  time  which  the  body  would  take  to  fall  to  the  deck, 
the  top  of  the  mast  would  move  from  A  to  C.  But  the  body, 
being  on  the  mast  at  the  moment  it  is  disengaged,  has  this 
motion  A  C  in  common  with  the  mast;  and,  therefore,  in  its 
descent  it  is  affected  by  two  motions,  viz.  that  of  the  vessel 
expressed  by  A  C,  and  its  descending  motion  expressed  by 
A  B.  Hence,  by  the  composition  of  motion,  it  will  be  found 
at  the  opposite  angle  D  of  the  parallelogram,  at  the  end. of 

5  » 


60  THE    ELEMENTS    OF   MECHANICS.  CHAP.    V 

the  fall.  During  the  fall,  however,  the  mast  has  moved  with 
the  vessel,  and  has  advanced  to  C  D,  so  that  the  body  falls  at 
the  foot  of  the  mast. 

(88.)  An  instance  of  the  composition  of  motion,  which  is 
worthy  of  some  attention,  as  it  affords  a  proof  of  the  diurnal 
motion  of  the  earth,  is  derived  from  observing  the  descent 
of  a  body  from  a  very  high  tower.  To  render  the  explana- 
tion of  this  more  simple,  we  shall  suppose  the  tower  to  be  on 
the  equator  of  the  earth.  Let  E  P  Q,Jig.  18.,  be  a  section 
of  the  earth  through  the  equator,  and  let  P  T  be  the  tower. 
Let  us  suppose  that  the  earth  moves  on  its  axis  in  the  direc- 
tion E  P  Q.  The  foot  P  of  the  tower  will,  therefore,  in  one 
day,  move  over  the  circle  E  P  Q,,  while  the  top  T  moves  over 
the  greater  circle  T  T'  R.  Hence  it  is  evident,  that  the  top 
of  the  tower  moves  with  greater  speed  than  the  foot,  and 
therefore  in  the  same  time  moves  through  a  greater  space. 
Now  suppose  a  body  placed  at  the  top  ;  it  participates  in  the  mo- 
tion which  the  top  of  the  tower  has  in  common  with  the  earth. 
If  it  be  disengaged,  it  also  receives  the  descending  motion 
T  P.  Let  us  suppose  that  the  body  would  take  five  sec- 
onds to  fall  from  T  to  P,  and  that  in  the  same  time  the  top 
T  is  moved  by  the  rotation  of  the  earth  from  T  to  T',  the 
foot  being  moved  from  P  to  P'.  The  falling  body  is  therefore 
endued  with  two  motions,  one  expressed  by  T  T',»  and  the 
other  by  T  P.  The  combined  effect  of  tltese  will  be  found 
in  the  usual  way  by  the  parallelogram.  Take  T  p  equal  to 
T  T',  the  body  will  move  from  T  top  in  the  time  of  the  fall, 
and  will  meet  the  ground  at  p.  But  since  T  T'  is  greater 
than  P  P7,  it  follows  that  the  pointy  must  be  at  a  distance 
from  P'  equal  to  the  excess  of  T  T'  above  P  P7.  Hence  the 
body  will  not  fall  exactly  at  the  foot  of  the  tower,  but  at  a 
certain  distance  from  it,  in  the  direction  of  the  earth's  mo- 
tion, that  is,  eastward.  This  is  found,  by  experiment,  to  be 
actually  the  case ;  and  the  distance  from  the  foot  of  the 
tower,  at  which  the  body  is  observed  to  fall,  agrees  with  that 
which  is  computed  from  the  motion  of  the  earth,  to  as  great 
a  degree  of  exactness  as  could  be  expected  from  the  nature 
of  the  experiment. 

(89.)  The  properties  of  compounded  motions  cause  some 
of  the  equestrian  feats  exhibited  at  public  spectacles  to  be 
performed  by  a  kind  of  exertion  very  different  from  that  the 
spectators  generally  attribute  to  the  performer.  For  exam- 
ple, the  horseman,  standing  on  the  saddle,  leaps  over  a  garter 


CHAP.    V.  FAMILIAR    EXAMPLES.  51 

extended  over  the  horse  at  right  angles  to  his  motion ;  the 
horse  passing  under  the  garter,  the  rider  lights  upon  the  sad- 
dle at  the  opposite  side.  The  exertion  of  the  performer,  in 
this  case,  is  not  that  which  he  would  use  were  he  to  leap 
from  the  ground  over  a  garter  at  the  same  height.  In  the 
latter  case,  he  would  make  an  exertion  to  rise,  and  at  the 
same  time  to  project  his  body  forward.  In  the  case,  however, 
of  the  horseman,  he  merely  makes  that  exertion  which  is 
necessary  to  rise  directly  upwards  to  a  sufficient  height  to 
clear  the  garter.  The  motion  which  he  has  in  common  with 
the  horse,  compounded  with  the  elevation  acquired  by  his 
muscular  power,  accomplishes  the  leap. 

To  explain  this  more  fully,  let  A  B  C,  j%.  19.,  be  the  di- 
rection in  which  the  horse  moves,  A  being  the  point  at  which 
the  rider  quits  the  saddle,  and  C  the  point  at  which  he 
returns  to  it.  Let  D  be  the  highest  point  which  is  to  be 
cleared  in  tha  leap.  At  A  the  rider  makes  a  leap  towards 
the  point  E,  and  this  must  be  done  at  such  a  distance  from  B, 
that  he  would  rise  from  B  to  E  in  the  time  in  which  the  horse 
moves  from  A  to  B.  On  departing  from  A,  the  rider  has, 
therefore,  two  motions,  represented  by  the  lines  A  E  and 
A  B,  by  which  he  will  move  from  the  point  A  to  the  opposite 
angle  D  of  the  parallelogram.  At  D,  the  exertion  of  the 
leap  being  overcome  by  the  weight  of  his  body,  he  begins  to 
return  downward,  and  would  fall  from  D  to  B  in  the  time  in 
which  the  horse  moves  from  B  to  C.  But  at  D  he  still 
retains  the  motion  which  he  had  in  common  with  the  horse : 
and,  therefore,  in  leaving  the  point  D,  he  has  two  motions, 
expressed  by  the  lines  D  F  and  D  B.  The  compounded 
effects  of  these  motions  carry  him  from  D  to  C.  Strictly 
speaking,  his  motion  from  A  to  D,  and  from  D  to  C,  is  not 
in  straight  lines,  but  in  a  curve.  It  is  not  necessary  here, 
however,  to  attend  to  this  circumstance. 

(90.)  If  a  billiard-ball  strike  the  cushion  of  the  table 
obliquely,  it  will  be  reflected  from  it  in  a  certain  direction, 
forming  an  angle  with  the  direction  in  which  it  struck  it. 
This  affords  an  example  of  the  resolution  and  composition  of 
motion.  We  shall  first  consider  the  effect  which  would 
ensue  if  the  ball  struck  the  cushion  perpendicularly. 

Let  A  B,  jig.  20.,  be  the  cushion,  and  C  D  the  direction 
in  which  the  ball  moves  towards  it.  If  the  ball  and  the 
cushion  were  perfectly  inelastic,  the  resistance  of  the  cushion 
would  destroy  the  motion  of  the  ball,  and  it  would  be  reduced 


5*4  THE  ELEMENTS  OP  MECHANICS.       CHAP.  V. 

to  a  state  of  rest  at  D.  If,  on  the  other  hand,  the  ball  were 
perfectly  elastic,  it  would  be  reflected  from  the  cushion,  and 
would  receive  as  much  motion  from  D  to  C,  after  the  im- 
pact, as  it  had  from  C  to  D  before  it.  Perfect  elasticity, 
however,  is  a  quality  which  is  never  found  in  these  bodies. 
They  are  always  elastic,  but  imperfectly  so.  Consequently, 
the  ball,  after  the  impact,  will  be  reflected  from  D  towards 

C,  but  with  a  less  motion  than  that  with  which  it  approach- 
ed from  C  to  D. 

Now  let  us  suppose  that  the  ball,  instead  of  moving  from 
C  to  D,  moves  from  E  to  D.  The  force  with  which  it  strikes 

D,  being  expressed  by  D  E',  equal  to  E  D,  may  be  resolved 
into  two,  D  F  and  D  C'.     The  resistance  of  the  cushion  de- 
stroys D  C7,  and  the  elasticity  produces  a  contrary  force  in 
the  direction  D  C,  but  less  than  D  C  or  D  C',  because  that 
elasticity  is  imperfect.     The  line  D  C  expressing  the  force 
in  the  direction  C  D,  let  D  G  (less  than  D  C)  express  the 
reflective  force  in  the  direction  D  C.     The  other  element, 
D  F,  into  which  the  force  D  E'  is  resolved  by  the  impact,  is 
not  destroyed  or  modified  by  the  cushion,  and  therefore,  on 
leaving  the  cushion  at.  D,  the  ball  is  influenced  by  two  forces, 
D   F  (which  is  equal  to  C  E)  and  D  G.     Consequently  it 
will  move  in  the  diagonal  D  H. 

(91.)  The  angle  E  D  C  is,  in  this  case,  called  the  "  angle 
of  incidence,"  and  C  D  H  is  called  "  the  angle  of  reflec- 
tion." It  is  evident,  from  what  has  been  just  inferred,  that, 
the  ball  being  imperfectly  elastic,  the  angle  of  incidence 
must  always  be  less  than  the  angle  of  reflection,  and,  with 
the  same  obliquity  of  incidence,  the  more  imperfect  the  elas- 
ticity is,  the  less  will  be  the  angle  of  reflection. 

In  the  impact  of  a  perfectly  elastic  body,  the  angle  of  re- 
flection would  be  equal  to  the  angle  of  incidence.  For  then 
the  line  D  G,  expressing  the  reflective  force,  would  be  taken 
equal  to  C  D,  and  the  angle  C  D  H  would  be  equal  to  C  D  E. 
This  is  found  by  experiment  to  be  the  case  when  light  is 
reflected  from  a  polished  surface  of  glass  or  metal. 

Motion  is  sometimes  distinguished  into  absolute  and  relative. 
What  "relative  motion"  means  is  easily  explained.  If  a 
man  walk  upon  the  deck  of  a  ship  from  stem  to  stern,  he 
has  a  relative  motion  which  is  measured  by  the  space  upon 
the  deck  over  which  he  walks  in  a  given  time.  But  while 
he  is  thus  walking  from  stem  to  stern,  the  ship  and  its  con- 
tents, including  himself,  are  impelled  through  the  deep  iu 


CHAP.  VI. 


ATTRACTION.  53 


the  opposite  direction.  If  it  so  happen  that  the  motion  of 
the  man  from  stem  to  stern  be  exactly  equal  to  the  motion 
of  the  ship  in  the  contrary  way,  the  man  will  be,  relatively 
to  the  surface  of  the  sea  and  that  of  the  earth,  at  rest.  Thus, 
relatively  to  the  ship,  he  is  in  motion,  while,  relatively  to  the 
surface  of  the  earth,  he  is  at  rest.  But  "still  this  is  not  abso- 
lute rest.  The  surface  itself  is  moving  by  the  diurnal  rota- 
tion of  the  earth  upon  its  axis,  as  well  as  by  the  animal 
motion  in  its  orbit  round  the  sun.  These  motions,  and 
others  to  which  the  earth  is  subject,  must  be  all  compounded 
by  the  theorem  of  the  parallelogram  of  forces,  before  we  can 
obtain  the  absolute  state  of  the  body  with  respect  to  motion 
or  rest. 


CHAPTER  VI. 

ATTRACTION. 

(92.)  WHATEVER  produces,  or  tends  to  produce,  a  change 
in  the  state  of  a  particle  or  mass  of  matter  with  respect  to 
motion  or  rest,  is  a  force.  Rest,  or  uniform  rectilinear  mo- 
tion, are  therefore  the  only  states  in  which  any  body  can 
exist  which  is  riot  subject  to  the  present  action  of  some  force. 
We  are  not,  however,  entitled  to  conclude,  that  because  a 
body  is  observed  in  one  or  other  of  these  states,  it  is  therefore 
uninfluenced  by  any  forces.  It  may  be  under  the  immedi- 
ate action  of  forces  which  neutralize  each  other ;  thus  two 
forces  may  be  acting  upon  it  which  are  equal,  and  in  oppo- 
site directions.  In  such  a  case,  its  state  of  rest,  or  of  uniform 
rectilinear  motion  will  be  undisturbed.  The  state  of  uni- 
form rectilinear  motion  declares  more  with  respect  to  the  body 
than  the  state  of  rest ;  for  the  former  betrays  the  action  of  a 
force  upon  the  body  at  some  antecedent  period  ;  this  action 
having  been  suspended,  while  .its  effect  continues  to  be  ob- 
served in  the  motion  which  it  has  produced. 

(93.)  When  the  state  of  a  body  is  changed  from  rest  to 
uniform  rectilinear  motion,  the  action  of  the  force  is  only 
momentary,  in  which  case  it  is  called  an  impulse.  If  a  body 
in  uniform  rectilinear  motion  receive  an  impulse  in  the  direc- 
tion in  which  it  is  moving,  the  effect  will  be,  that  it  will 
continue  to  move  uniformly  in  the  same  direction,  but  its 


54  THE  ELEMENTS   OF  MECHANICS.  CHAP.   VI 

velocity  will  be  increased  by  the  amount  of  speed  which 
the  impulse  would  have  given  it,  had  it  been  previously 
quiescent.  Thus,  if  the  previous  motion  be  at  the  rate  of  ten 
feet  in  a  second,  and  the  impulse  be  such  as  would  move  it 
from  a  state  of  res.t  at  five  feet  in  a  second,  the  velocity, 
after  the  impulse,  will  be  fifteen  feet  in  a  second. 

But  if  the  impulse  be  received  in  a  direction  immediately 
opposed  to  the  previous  motion,  then  it  will  diminish  the 
speed  by  that  amount  of  velocity  which  it  would  give  to  the 
body  had  it  been  previously  at  rest.  In  the  example  already 
given,  if  the  impulse  were  opposed  to  the  previous  motion, 
the  velocity  of  the  body  after  the  impulse  would  be  five  feet 
in  a  second.  If  the  impulse  received  in  the  direction  opposed 
to  the  motion  be  such  as  would  give  to  the  body  at  rest  a 
velocity  equal  to  that  with  which  it  is  moving,  then  the  effect  will 
be,  that  after  the  impulse  no  motion  will  exist ;  and  if  the 
impulse  would  give  it  a  still  greater  velocity,  the  body  will  be 
moved  in  the  opposite  direction  with  an  uniform  velocity 
equal  to  the  excess  of  that  due  to  the  impulse  over  that  which 
the  body  previously  had. 

When  a  body  in  a  state  of  uniform  motion  receives  an 
impulse  in  a  direction  not.  coinciding  with  that  of  its  motion, 
it  will  move  uniformly,  after  the  impulse,  in  an  intermediate 
direction,  which  may  be  determined  by  the  principles  estab- 
lished for  the  composition  of  motion  in  the  last  chapter. 

Thus  it.  appears,  that  whenever  the  state  of  a  body  is 
changed,  either  from  rest  to  uniform  rectilinear  motion,  or 
rice,  versa,  or  from  one  state  of  uniform  rectilinear  motion 
to  another,  differng  from  that,  eithjer  in  velocity  or  direction, 
or  in  both,  the  phenomenon  is  produced  by  that  peculiar 
modification  of  force  whose  action  continues  but  for  a  single 
instant,  and  which  has  been  called  an  impulse. 

(94.)  In  most  cases,  however,  the  mechanical  state  of  a 
body  is  observed  to  be  subject  to  a  continual  change  or  ten- 
dency to  change.  We  are  surrounded  by  innumerable  ex- 
amples of  this.  A  body  is  placed  on  the  table.  A  continual 
pressure  is  excited  on  the  surface  of  the  table.  This  pressure 
is  only  the  consequence  of  the  continual  tendency  of  the 
body  to  move  downwards.  If  the  body  were  excited  by  a 
force  of  the  nature  of  an  impulse,  the  effect  upon  the  table- 
would  be  instantaneous,  and  would  immediately  cease.  It 
would,  in  fact,  be  a  blow.  But  the  continuation  of  the  pres- 
sure proves  the  continuation  of  the  action  of  the  force. 


CHAP.  VI.  ATTRACTION,  55 

If  the  table  he  removed  from  beneath  the  body,  the  force 
which  excites  it,  being  no  longer  resisted,  will  produce  motion; 
it  is  manifested,  not  as  before,  by  a  tendency  to  produce 
motion,  but  by  the  actual  exhibition  of  that  phenomenon. 
Now,  if  the  exciting  force  were  an  impulse,  the  body  would 
descend  to  the  ground  with  an  uniform  velocity.  On  the 
other  hand,  as  will  hereafter  appear,  every  moment  of  its  fall 
increases  its  speed,  and  that  speed  is  greatest  at  the  instant 
it  meets  the  ground. 

A  piece  of  iron  placed  at  a  distance  from  a  magnet  ap- 
proaches it,  but  not  with  an  uniform  velocity.  The  force 
of  the  magnet  continues  to  act.  during  the  approach  of  the 
iron,  and  each  moment  gives  it  increased  motion. 

(95.)  The  forces  which  are  thus  in  constant  operation, 
proceed  from  secret  agencies  which  the  human  mind  has 
novcr  !»:vm  able  to  detect.  All  the  analogies  of  nature  prove 
that  they  are  not  the  immediate  results  of  the  divine  will, 
but  are  secondary  causes,  that  is,  effects  of  some  more  remote 
principles.  To  ascend  to  these  secondary  causes,  and  thus, 
as  it  were,  approach  one  step  nearer  to  the  Creator,  is  the 
great  business  of  philosophy  ;  and  the  most  certain  means 
ior  accomplishing  this,  is  diligently  to  observe,  to  compare, 
and  to  classify  the  phenomena,  and  to  avoid  assuming  the  ex- 
istence of  any  thing  which  has  not  either  been  directly  ob- 
served, or  which  cannot  be  inferred  demonstratively  from 
natural  phenomena.  Philosophy  should  follow  nature,  and 
not  lead  her. 

While  the  law  of  inertia,  established  by  observation  and 
reason,  declares  the  inability  of  matter,  from  any  principle 
resident  in  it,  to  change  its  state,  all  the  phenomena  of  the 
universe  prove  that  state  to  be  in  constant  but  regular  fluc- 
tuation. There  is  not  in  existence  a  single  instance  of  the 
phenomenon  of  absolute  rest,  or  of  motion  which  is  absolutely 
uniform  and  rectilinear.  In  bodies,  or  the  parts  of  bodies, 
there  is  no  known  instance  of  simple  passive  juxtaposition 
unaccompanied  by  pressure  or  tension,  or  some  other  "  ten- 
dency to  motion."  Innumerable  secret  powers  are  ever  at 
work,  compensating,  as  it  were,  for  inertia,  and  supplying 
the  material  world  with  a  substitute  for  the  principles  of 
action  and  will,  which  give  such  immeasurable  superiority  to 
the  character  of  life. 

(9(3.)  The  forces  which  are  thus  in  continual  operation, 
whose  existence  is  demonstrated  by  their  observed  effects, 


56  THE    ELEMENTS    OF    MECHANICS.  CHAP.  VI. 

but  whose  nature,  seat,  and  mode  of  operation,  are  unknown 
to  us,  are  called  by  the  general  name  attractions.  These 
forces  are  classified  according  to  the  analogies  which  prevail 
among  their  effects,  in  the  same  manner,  and  according  to 
the  same  principles,  as  organized  beings  are  grouped  in  natu- 
ral history.  In  that  department  of  natural  science,  when 
individuals  are  distributed  in  classes,  the  object  is  merely 
to  generalize,  and  thereby  promote  the  enlargement  of  knowl- 
edge ;  but  nothing  is  or  ought  to  be  thus  assumed  respecting 
the  essence,  or  real  internal  constitution  of  the  individuals. 
According  to  their  external  and  observable  characters  and 
qualities  they  are  classed;  aud  this  classification  should  never 
be  adduced  as  an  evidence  of  any  thing  except  that  similitude 
of  qualities  to  which  it  owed  its  origin. 

Phenomena  are  to  the  natural  philosopher  what  organized 
beings  are  to  the  naturalist.  He  groups  and  classifies  them 
on  the  same  principles,  and  with  a  like  object.  And  as  the. 
naturalist  gives  to  each  species  a  name  applicable  to  the 
individual  beings  which  exhibit  corresponding  qualities,  so 
the  philosopher  gives  to  each  force  or  attraction  a  name  cor- 
responding to  the  phenomena  of  wliicb  it  is  the  cause.  The 
naturalist  is  ignorant  of  the  real  essence  or  internal  constitu- 
tion of  the  thing  which  he  nominates,  and  of  lae  manner  in 
which  it  comes  to  possess  or  exhibit  those  qualities  which 
form  the  basis  of  his  classification ;  and  the  natural  philoso- 
pher is  equally  ignorant  of  the  nature,  seat,  and  mode  of 
operation  of  the  force  which  he  assigns  as  the  cause  of  an 
observed  class  of  effects. 

These  observations  respecting  the  true  import  of  the  term 
"  attraction"  seem  the  more  necessary  to  be  premised,  be- 
cause the  general  phraseology  of  physical  science,  taken  as 
language  is  commonly  received,  will  seem  to  convey  some- 
thing more.  The  names  of  the  several  attractions  which  we 
shall  have  to  notice,  frequently  refer  the  seat  of  the  cause 
to  specific  objects,  and  seem  to  imply  something  respecting 
its  mode  of  operation.  Thus,  when  we  say,  "  the  magnet 
attracts  a  piece  of  iron,"  the  true  philosophical  import  of  the 
words  is,  "  that  a  piece  of  iron,  placed  in  the  vicinity  of  the 
magnet,  will  move  towards  it,  or,  placed  in  contact,  will 
adhere  to  it,  so  that  some  force  is  necessary  to  separate  them." 
In  the  ordinary  sense,  however,  something  more  than  this 
simple  fact  is  implied.  It  is  insinuated  that  the  magnet  is 
the  seat  of  the  force  which  gives  motion  to  the  iron  ;  that, 


CILVP.  Vf.  ATTRACTION.  57 

in  the  production  of  the  phenomenon,  the  magnet  is  an  agent 
exerting  a  certain  influence,  of  which  the  iron  is  the  subject. 
Of  all  this,  however,  there  is  no  proof;  on  the  contrary,  since 
the  magnet  must  move  towards  the  iron  with  just  as  much 
force  as  the  iron  moves  towards  the  magnet,  there  is  as  much 
reason  to  place  the  seat  of  the  force  in  the  iron,  and  consider 
it  as  an  ^igent  affecting  the  magnet.  But,  in  fact,  the  influ- 
ence which  produces  this  phenomenon  may  not  be  resident 
in  either  the  one  body  or  the  other.  It  may  be  imagined  to 
be  a  property  of  a  medium  in  which  both  are  placed,  or  to 
arise  from  some  third  body,  the  presence  of  which  is  not  im- 
mediately observed.  However  attractive  these  and  like  spec- 
ulations may  be,  they  cannot  be  allowed  a  place  in  physical 
investigations,  nor  should  consequences  drawn  from  such 
hypotheses  be  allowed  to  taint  our  conclusions  with  their  un- 
certainty. 

The  student  ought,  therefore,  to  be  aware,  that  whatever 
may  seem  to  be  implied  by  the  language  used  in  this  science 
in  relation  to  attractions,  nothing  is  permitted  to  form  the 
basis  of  reasoning  respecting  them  except  their  effects  ;  and 
whatever  be  the  common  signification  of  the  terms  used, 
it  is  to  these  effects,  and  to  these  alone,  they  should  be  re- 
ferred. 

(97.)  Attractions  may  be  primarily  distributed  into  two 
classes  ;  one  consisting  of  those  which  exist  between  the 
molecules  or  constituent  parts  of  bodies,  and  the  other  be- 
tween bodies  themselves.  The  former  are  sometimes  called, 
for  distinction,  molc.c.ular  or  atomic  attractions. 

Without  the  agency  of  molecular  forces,  the  whole  face 
of  nature  would  be  deprived  of  variety  and  beauty  ;  the  uni- 
verse would  be  a  confused  heap  of  material  atoms  dispersed 
through  spa*-e,  without  form,  shape,  coherence,  or  motion. 
Bodies  would  neither  have  the  forms  of  solid,  liquid,  or  air ; 
heat  and  light  would  no  longer  produce  their  wonted  effects  ; 
organized  beings  could  not  exist ;  life  itself,  as  connected 
with  body,  would  be  extinct.  Atoms  of  matter,  whether  dis- 
tant or  in  juxtaposition,  would  have  no  tendency  to  change 
their  places,  and  all  would  be  eternal  stillness  and  rest.  If, 
then,  we  are  asked  for  a  proof  of  the  existence  of  molecular 
forces,  we  may  point  to  the  earth  and  to  the  heavens  ;  we 
may  name  every  object  which  can  be  seen  or  felt.  The 
whole  material  world  is  one  great  result  of  the  influence  of 
these  powerful  agents. 


58  THE  ELEMENTS  OF  MECHANICS.       CHAP.  VI. 

(98.)  It  has  been  proved  (11.  et  srq.)  that  the  constituent 
particles  of  bodies  are  of  inconceivable  minuteness,  and  that 
they  are  not  in  immediate  contact  (26),  but  separated  from 
each  other  by  interstitial  spaces,  which,  like  the  atoms  them- 
selves, although  too  small  to  be  directly  observed,  yet  are 
incontestably  proved  to  exist,  by  observable  phenomena,  from 
which  their  existence  demonstratively  follows.  The  resist- 
ance which  every  body  opposes  to  compression,  proves  that  a 
repulsive  influence  prevails  between  the  particles,  and*  that 
this  repulsion  is  the  cause  which  keeps  the  atoms  separate, 
and  maintains  the  interstitial  space  just  mentioned.  Although 
this  repulsion  is  found  to  exist  between  the  molecules  of  all 
substances  whatever,  yet  it  has  different  degrees  of  energy  in 
different  bodies.  This  is  proved  by  the  fact,  that  some  sub- 
stances admit  of  easy  compression,  while,  in  others,  the  exer- 
tion of  considerable  force  is  necessary  to  produce  the  smallest 
diminution  in  bulk. 

The  space  around  each  atom  of  a  body,  through  which 
this  repulsive  influence  extends,  is  generally  limited,  and 
immediately  beyond  it,  a  force  of  the  opposite  kind  is  mani- 
fested, viz.  attraction.  Thus,  in  solid  bodies,  the  particles 
resist  separation  as  well  as  compression,  and  the  application 
of  force  is  as  necessary  to  break  the  body,  or  divide  it  into 
separate  parts,  as  to  force  its  particles  into  closer  aggregation. 
It  is  by  virtue  of  this  attraction  that  solid  bodies  maintain 
their  figure,  and  that  their  parts  are  not  separated  and  scat- 
tered like  those  of  fluids,  merely  by  their  own  weight.  This 
force  is  called  the  attraction  of  cohesion. 

The  cohesive  force  acts  in  different  substances  with  differ- 
out  degrees  of  energy  :  in  some  its  intensity  is  very  great, 
but  the  sphere  of  its  influence  apparently  very  limited.  This 
is  the  case  with  all  bodies  which  are  hard,  strong,  and  brittle, 
which  no  force  can  extend  or  stretch  in  any  perceptible  de- 
gree, and  which  require  a  great  force  to  break  or  tear  them 
usunder.  Such,  for  example,  is  cast  iron,  certain  stones, 
and  various  other  substances.  In  some  bodies,  the  cohesive 
force  is  weak,  but  the  sphere  of  its  action  considerable.  Bod- 
ies which  are  easily  extended,  without  being  broken  or  torn 
asunder,  furnish  examples  of  this.  Such  are  Indian-rubber, 
or  caoutchouc,  several  animal  and  vegetable  products,  and, 
in  general,  all  solids  of  a  soft  and  viscid  kind. 

Between  these  extremes,  the  cohesive  force  may  be  ob- 
served in  various  decrees.  In  lead  and  other  soft  metals, 


CHAP.  VI.  COHESION.  59 

its  sphere  of  action  is  greater,  and  its  energy  less,  than  in  the 
former  examples  ;  but  its  sphere  less,  and  energy  greater, 
than  in  the  latter  ones.  It  is  from  the  influence  of  this  force, 
and  that  of  the  repulsion,  whose  sphere  of  action  is  still  closer 
to  the  component  atoms,  that  all  the  varieties  of  form  which 
we  denominate  hard,  soft,  tough,  brittle,  ductile,  pliant,  &/c. 
arise. 

After  having  been  broken,  or  otherwise  separated,  the 
parts  of  a  solid  may  bo  again  united  by  their  cohesion,  pro- 
vided any  considerable  number  of  points  be  brought  into  suf- 
ficiently close  contact.  When  this  is  done  by  mechanical 
means,  however,  the  cohesion  is  not  so  strong  as  before  their 
separation,  and  a  comparatively  small  force  will  be  sufficient 
again  to  disunite  them.  Two  pieces  of  lead  freshly  cut,  with 
smooth  surfaces,  will  adhere  when  pressed  together,  and  will 
require  a  considerable  force  to  separate  them.  In  the  same 
manner,  if  a  piece  of  Indian-rubber  be  torn,  the  parts  sepa- 
rated will  again  cohere,  by  being  brought  together  with  a 
slight  pressure.  The  union  of  the  parts,  in  such  instances, 
is  easy,  because  the  sphere  through  which  the  influence  of 
cohesion  extends  is  considerable  ;  but  even  in  bodies  in  which 
this  influence  extends  through  a  more  limited  space,  the  co- 
hesion of  separate  pieces  will  be  manifested,  provided  their 
surfaces  be  highly  polished,  so  as  to  insure  the  near  approach 
of  a  great  number  of  their  particles.  Thus  two  polished 
surfaces  of  glass,  metal,  or  stone,  will  adhere  when  brought 
into  contact. 

In  all  these  cases,  if  the  bodies  be  disunited  by  mechanical 
force,  they  will  separate  at  exactly  the  parts  at  which  they 
had  been  united,  so  that,  after  their  separation,  no  part  of  the 
one  will  adhere  to  the  other  ;  proving  that  the  force  of  cohe- 
sion of  the  surfaces  brought  into  contact  is  less  than  that 
which  naturally  held  the  particles  of  each  together. 

(99.)  When  a  body  is  in  the  liquid  form,  the  weight  of  its 
particles  greatly  predominates  over  their  mutual  cohesion, 
and,  consequently,  if  such  a  body  be  uncontined,  it  will  be 
scattered  by  its  own  weight ;  if  it  be  placed  in  any  vessel, 
it  will  settle  itself,  by  the  force  of  its  weight,  into  the  lowest 
parts,  so  that  no  space  in  the  vessel  below  the  upper  surface 
of  the  liquid  will  be  unoccupied.  The  particles  of  a  solid 
body  placed  in  the  vessel  have  exactly  the  same  tendency, 
by  reason  of  their  weight ;  but  this  tendency  is  resisted  and 
prevented  from  taking  effect  by  their  strong  cohesion. 


60  THE  ELEMENTS  OF  MECHANICS.  CHAP.  VI 

Although  this  cohesion  in  solids  is  much  greater  than  in 
liquids,  and  productive  of  more  obvious  effects,  yet  the  prin- 
ciple is  not  altogether  unobserved  in  liquids.  Water  convert- 
ed into  vapor  by  heat,  is  divided  into  inconceivably  minute 
particles,  which  ascend  in  the  atmosphere.  When  it  is  there 
deprived  of  a  part  of  that  heat  which  gave  it  the  vaporous 
form,  the  particles,  in  virtue  of  their  cohesive  force,  collect 
into  round  drops,  in  which  form  they  descend  to  the  earth. 

In  the  same  manner,  if  a  liquid  be  allowed  to  fall  gradu- 
ally from  the  lip  of  a  vessel,  it  will  not  be  dismissed  in  parti- 
cles indefinitely  small,  as  if  its  mass  were  incoherent,  like 
sand  or  powder,  but  will  fall  in  drops  of  considerable  magni- 
tude. In  proportion  as  the  cohesive  force  is  greater,  these 
drops  affect  a  greater  size.  Thus,  oil  and  viscid  liquids  fall 
in  large  drops  ;  ether,  alcohol,  and  others,  in  small  ones. 

Two  drops  of  rain  trickling  down  a  window  pane  will 
coalesce  when  they  approach  each  other  ;  and  the  same  phe- 
nomenon is  still  more  remarkable,  if  a  few  drops  of  quick- 
silver be  scattered  on  an  horizontal  plate  of  glass. 

It  is  the  cohesive  principle  which  gives  rotundity  to  grains 
of  shot :  the  liquid  metal  is  allowed  to  fall  like  rain  from  a 
great  elevation.  In  its  descent,  the  drops  become  truly  glob- 
ular, and  before  they  reach  the  end  of  their  fall,  they  are 
hardened  by  cooling,  so  that  they  retain  their  shape. 

It  is  also,  probably,  to  the  cohesive  attraction  that  we 
should  assign  the  globular  forms  of  all  the  great  bodies  of 
the  universe  ;  the  sun,  planets,  satellites,  &c.,  which  origi- 
nally may  have  been  in  the  liquid  state. 

(100.)  Molecular  attraction  is  also  exhibited  between  the 
particles  of  liquids  and  solids.  A  drop  of  water  will  not 
descend  freely  when  it  is  in  contact  with  a  perpendicular 
glass  plane :  it  will  adhere  to  the  glass ;  its  descent  will  be 
retarded  ;  and  if  its  weight  be  insufficient  to  overcome  the 
adhesive  force,  it  will  remain  suspended. 

If  a  plate  of  glass  be  placed  upon  the  surface  of  water 
without  being  permitted  to  sink,  it  will  require  more  force  to 
raise  it  from  the  water  than  is  sufficient  merely  to  balance  the 
weight  of  the  glass.  This  shows  the  adhesion  of  the  water 
and  glass,  and  also  the  cohesive  force  with  which  the  particles 
of  the  water  resist  separation. 

If  a  needle  be  dipped  in  certain  liquids,  a  drop  will  remain 
suspended  at  its  point  when  withdrawn  from  them :  and,  in 
general,  when  a  solid  body  has  been  immersed  in  a  liquid,  and 


CHAP.  VI.         MOLECULAR  ATTRACTION.  61 

withdrawn,  it  is  wet;  that  is,  some  of  the  liquid  has  adhered 
to  its  surfaces.  If  no  attraction  existed  between  the  solid 
and  liquid,  the  solid  would  be  in  the  same  state  after  immer- 
sion as  before.  This  is  proved  by  liquids  and  solids  between 
which  no  attraction  exists.  If  a  piece  of  glass  be  immersed 
in  mercury,  it  will  be  in  the  same  state  when  withdrawn  as 
before  it  was  immersed.  No  mercury  will  adhere  to  it ;  it  will 
not  be  wet. 

When  it  rains,  the  person  and  vesture  are  affected  only  be- 
cause this  attraction  exists  between  them  and  water.  If  it 
rained  mercury,  none  would  adhere  to  them. 

(101.)  When  molecular  attraction  is  exhibited  by  liquids 
pervading  the  interstices  of  porous  bodies,  ascending  in  crev- 
ices or  in  the  bores  of  small  tubes,  it  is  called  capillary  at- 
traction. Instances  of  this  are  innumerable.  Liquids  are 
thus  drawn  into  the  pores  of  sponge,  sugar,  lamp-wick,  &c. 
The  animal  and  vegetable  kingdom  furnish  numerous  exam- 
ples of  this  class  of  effects. 

A  weight,  being  suspended  by  a  dry  rope,  will  be  drawn 
upwards  through  a  considerable  height,  if  the  rope  be  moist- 
ened with  a  wet  sponge.  The  attraction  of  the  particles  com- 
posing the  rope  for  those  of  the  water  is  in  this  case  so  power- 
ful, that  the  tension  produced  by  several  hundred  weight  can- 
not expel  them. 

A  glass  tube,  of  small  bore,  being  dipped  in  water  tinged 
by  mixture  with  a  little  ink,  will  retain  a  quantity  of  the  liquid 
suspended  when  withdrawn.  The  height  of  the  liquid  in  the 
tube  will  be  seen  by  looking  through  it.  It  is  found  that  the 
less  the  bore  of  the  tube  is,  the  greater  will  be  the  height  of 
the  column  sustained.  A  series  of  such  tubes  fixed  in  the 
same  frame,  with  their  lower  orifices  at  the  same  level,  and 
with  bores  gradually  decreasing,  being  dipped  in  the  liquid, 
will  exhibit  columns  gradually  increasing. 

A  capillary  syphon  is  formed  of  a  hank  of  cotton  threads, 
one  end  of  which  is  immersed  in  the  vessel  containing  the 
liquid,  and  the  other  is  carried  into  the  vessel  into  which  the 
liquid  is  to  be  transferred.  The  liquid  may  be  thus  drawn 
from  the  one  vessel  into  the  other.  The  same  effect  may  be 
produced  by  a  glass  syphon  with  a  small  bore. 

(102.)  It  frequently  happens  that  a  molecular  repulsion  is 

exhibited  between  a  solid  and  a  liquid.     If  a  piece  of  wood 

be  immersed  in  quicksilver,  the  liquid  will  be  depressed  at 

that  part  of  the  surface  which  is  near  the  wood  ;  and  in  like 

6 


62  THE    ELEMENTS    OF    MECHANICS.  CHAP.    VI. 

manner,  if  it  be  contained  in  a  glass  vessel,  it  will  be  depress- 
ed at  the  edges.  In  a  barometer  tube,  the  surface  of  the 
mercury  is  convex,  owing  partly  to  the  repulsion  between  the 
glass  and  mercury. 

All  solids,  however,  do  not  repel  mercury.  If  any  golden 
trinket  be  dipped  in  that  liquid,  or  even  be  exposed  for  a  mo- 
ment to  contact  with  it,  the  gold  will  be  instantly  intermingled 
with  particles  of  quicksilver,  the  metal  changes  its  color,  and 
becomes  white  like  silver,  and  the  mercury  can  only  be  extri- 
cated by  a  difficult  process.  Chains,  seals,  rings,  &,c.,  should 
always  be  laid  aside  by  those  engaged  in  experiments  or  other 
processes  in  which  mercury  is  used. 

(103.)  Of  all  the  forms  under  which  molecular  force  is  ex- 
hibited, that  in  which  it  takes  the  name  of  affinity  is  attend- 
ed with  the  most  conspicuous  eifects.  Affinity  is  in  chemis- 
try what  inertia  is  in  mechanics — the  basis  of  the  science. 
The  present  treatise  is  not  the  proper  place  for  any  detailed 
account  of  this  important  class  of  natural  phenomena.  Those 
who  seek  such  knowledge  are  referred  to  our  treatise  on 
CHEMISTRY.  Since,  however,  affinity  sometimes  influences 
the  mechanical  state  of  bodies,  and  affects  their  mechanical 
properties,  it  will  be  necessary  here  to  state  so  much  respect- 
ing it  as  to  render  intelligible  those  references  which  we  may 
have  occasion  to  make  to  such  effects. 

When  the  particles  of  different  bodies  are  brought  intc 
close  contact,  and  more  especially  when,  being  in  a  fluid 
state,  they  are  mixed  together,  their  union  is  frequently  ob- 
served to  produce  a  compound  body,  differing  in  its  qualities 
from  either  of  the  component  bodies.  Thus  the  bulk  of  the 
compound  is  often  greater  or  less  than  the  united  volumes  of 
the  component  bodies.  The  component  bodies  may  be  of 
the  ordinary  temperature  of  the  atmosphere,  and  yet  the  com- 
pound may  be  of  a  much  higher  or  lower  temperature.  The 
components  may  be  liquid,  and  the  compound  solid.  The 
color  of  the  compound  may  bear  no  resemblance  whatever  to 
that  of  the  components.  The  species  of  molecular  action  be- 
tween the  component?:,  which  produce  these  and  similar  ef- 
fects, is  called  affinity. 

(104.)  We  shall  limit  ourselves  here  to  the  statement  of  a 
few  examples  of  these  phenomena. 

If  a  pint  of  water  and  a  pint  of  sulphuric  acid  be  mixed, 
the  compound  will  be  considerably  less  than  a  quart.  The 
density  of  the  mixture  is,  therefore,  greater  than  that  which 


CHAP.    VI.  AFFINITY.  63 

would  result  from  the  mere  diffusion  of  the  particles  of  the  one 
fluid  through  those  of  the  other.  The  particles  have  as- 
sumed a  greater  proximity,  and  therefore  exhibit  a  mutual 
attraction. 

In  this  experiment,  although  the  liquids  before  being  mixed 
be  of  the  temperature  of  the  surrounding  air,  the  mixture  will 
be  so  intensely  hot,  that  the  vessel  which  contains  it  cannot 
be  touched  without  pain. 

If  the  two  aeriform  fluids,  called  oxygen  and  hydrogen,  be 
mixed  together  in  a  certain  proportion,  the  compound  will  be 
water.  In  this  case,  the  components  are  different  from  the 
compound,  not  merely  in  the  one  being  air  and  the  other 
liquid,  but  in  other  respects  not  less  striking.  The  com- 
pound, water,  extinguishes  fire,  and  yet  of  the  components, 
hydrogen  is  one  of  the  most  inflammable  substances  in  nature, 
and  the  presence  of  oxygen  is  indispensably  necessary  to  sus- 
tain the  phenomenon  of  combustion. 

Oxygen  gas,  united  with  quicksilver,  produces  a  compound 
of  a  black  color,  the  quicksilver  being  white  and  the  gas 
colorless.  When  these  substances  are  combined  in  another 
proportion,  they  give  a  red  compound. 

(105.)  Having  noticed  the  principal  molecular  forces,  we 
(shall  now  proceed  to  the  consideration  of  those  attractions 
which  are  exhibited  between  bodies  existing  in  masses.  The 
influence  of  molecular  attractions  is  limited  to  insensible  dis- 
tances. On  the  contrary,  the  forces  which  are  now  to  be 
noticed,  act  at  considerable  distances,  and  to  the  influence 
of  some  there  is  no  limit,  the  effect,  however,  decreasing  as 
the  distance  increases. 

The  effect  of  the  loadstone  on  iron  is  well  known,  and  is 
one  of  this  class  of  forces.  For  a  detailed  account  of  this 
force,  and  the  various  phenomena  of  which  it  is  the  cause, 
the  reader  is  referred  to  our  treatise  on  MAGNETISM. 

When  glass,  wax,  amber,  and  other  substances,  are  submit- 
ted to  friction  with  silken  or  woollen  cloth,  they  are  observed 
to  attract  feathers,  and  other  light  bodies  placed  near  them. 
A  like  effect  is  produced  in  several  other  ways,  and  is  attend- 
ed with  other  phenomena,  the  discussion  of  which  forms  a 
principal  part  of  physical  science.  The  force  thus  exhib- 
ited is  called  electricity.  For  details  respecting  it,  and  for 
its  connection  with  magnetism,  the  reader  is  referred  to  our 
treatises  on  ELECTRICITY  and  ELECTRO-MAGNETISM. 

M06.)  These  attractions  exist  either   between  bodies  of 


64  THE    ELEMENTS    OF    MECHANICS.  CHAP.    VJ. 

particular  kinds,  or  are  developed  by  reducing  the  bodies 
which  manifest  them  to  a  certain  state  by  friction,  or  some 
other  means.  There  is,  however,  an  attraction,  which  is 
manifested  between  bodies  of  all  species,  and  under  all  cir- 
cumstances whatever ;  an  attraction,  the  intensity  of  which 
is  wholly  independent  of  the  nature  of  the  bodies,  and  only 
depends  on  their  masses  and  mutual  distances.  Thus,  if  a 
mass  of  metal  and  a  mass  of  clay.be  placed  in  the  vast  abyss 
of  space,  at  a  mile  asunder,  they  will  instantly  commence  to 
approach  each  other  with  certain  velocities.  Again,  if  a  mass 
of  stone  and  of  wood  respectively  equal  to  the  former,  be 
placed  at  a  like  distance,  they  will  also  commence  to  approach 
each  other  with  the  same  velocities  as  the  former.  This 
universal  attraction,  which  only  depends  on  the  quantity  of 
the  masses  and  their  mutual  distances,  is  called  the  "  attrac- 
tion of  gravitation."  We  shall  first  explain  the  "law"  of  this 
attraction,  and  shall  then  point  out  some  of  the  principal  phe- 
nomena by  which  its  existence  and  its  law  are  known. 

(107.)  The  "law  of  gravitation,"  sometimes,  from  its  uni- 
versality, called  the  "  law  of  nature,"  rnay  be  explained  as 
follows : 

Let  us  suppose  two  masses,  A  and  B,  in  pure  space,  beyond 
the  influence  or  attraction  of  any  other  bodies,  and  placed  in 
a  state  of  rest,  at  any  proposed  distance  from  each  other. 
By  their  mutual  attraction  they  will  approach  each  other,  but 
not  with  the  same  velocity.  The  velocity  of  A  will  be  great- 
er than  that  of  B,  in  the  same  proportion  as  its  mass  is  less 
than  that  of  B.  Thus,  if  the  mass  of  B  be  twice  that  of  A, 
while  A  approaches  B  through  a  space  of  two  feet,  B  will 
approach  A  through  a  space  of  one  foot.  Hence  it  follows, 
that  the  force  with  which  A  moves  towards  B  is  equal  to  the 
force  with  which  B  moves  towards  A  (f>8).  This  is  only  a 
consequence  of  the  property  of  inertia,  and  is  an  example  of 
the  equality  of  action  and  reaction,  as  explained  in  Chapter 
IV.  The  velocity  with  which  A  and  B  approach  each  other 
is  estimated  by  the  diminution  of  their  distance,  A  B,  by  their 
mutual  approach  in  a  given  time.  Thus,  if  in  one  second  A 
move  towards  B  through  a  space  of  two  feet,  and  in  the  same 
time  B  move  towards  A  through  the  space  of  one  foot,  they 
will  approach  each  other  through  a  space  of  three  feet  in  a 
second,  which  will  be  their  relative  velocity  (01). 

If  the  mass  of  B  be  doubled,  it  will  attract  A  with  double 
the  former  force,  or,  what  is  the  same,  will  cause  A  to  ap- 


CHAP.    VI.  GRAVITATION.  65 

proach  B  with  double  the  former  velocity.  If  the  mass  of  B 
be  trebled,  it  will  attract  A  with  treble  the  first  force,  and,  in 
general,  while  the  distance  A  B  remains  the  same,  the  attrac- 
tive force  of  B  upon  A  will  increase  or  diminish  in  exactly 
the  same  proportion  as  the  mass  of  B  is  increased  or  dimin- 
ished. 

In  the  same  manner,  if  the  mass  A  be  doubled,  it  will  be 
attracted  by  B  with  a  double  force,  because  B  exerts  the  same 
degree  of  attraction  on  every  part  of  the  mass  A,  arid  any 
addition  which  it  may  receive  will  not  diminish  or  otherwise 
affect  the  influence  of  B  on  its  former  mass. 

Thus  it  is  a  general  law  of  gravitation,  that  so  long  as  the 
distance  between  two  bodies  remains  the  same,  each  will  at- 
tract and  be  attracted  by  the  other,  in  proportion  to  its  mass  ; 
and  any  increase  or  decrease  of  the  mass  will  cause  a  corre- 
sponding increase  or  decrease  in  the  amount  of  the  attraction. 

(108.)  We  shall  now  explain  the  law,  according  to  which 
the  attraction  is  changed,  by  changing  the  distance  between 
the  bodies.  At  the  distance  of  one  mile,  the  body  B  attracts 
A  with  a  certain  force.  At  the  distance  of  two  miles,  the 
masses  not  being  changed,  the  attraction  of  B  upon  A  will 
be  one  fourth  of  its  amount  at  the  distance  of  one  mile.  At 
the  distance  of  three  miles,  it  will  be  one  ninth  of  its  original 
amount;  at  four  miles,  it  is  reduced  to  a  sixteenth,  and  so 
on.  The  following  table  exhibits  the  diminution  of  the  at- 
traction corresponding  to  the  successive  increase  of  distance  : 


{Distance 

1 

1    ^ 

I  » 

1    4    | 

5 

1  « 

1  7 

1  » 

|  &,c. 

Attraction 

1 

l  * 

l  * 

IA  1 

A 

1  A 

IA 

IA 

|  &.C. 

In  ARITHMETIC,  that  number  which  is  found  by  multiplying 
any  proposed  number  by  itself,  is  called  its  square.  Thus  4, 
that  is,  2  multiplied  by  2,  is  the  square  of  2 ;  9,  that  is,  3 
times  3,  is  the  square  of  3 ;  and  so  on.  On  inspecting  the 
above  table,  it  will  be  apparent,  therefore,  that  the  attraction 
of  gravitation  decreases  in  the  same  proportion  as  the  square 
of  the  distance  from  the  attracting  body  increases,  the  mass 
of  both  bodies  in  this  case  being  supposed  to  remain  the 
same;  but  if  the  mass  of  either  be  increased  or  diminished, 
the  attraction  will  be  increased  or  diminished  in  the  same 
proportion. 

(109.)  Hence  the  low  of  nature  may  be  thus  expressed  : 
"  The  mutual  attraction  of  two  bodies  increases  in  thft  same* 
C* 


THE    ELEMENTS    OF    MECHANICS.  CHAP.    VI. 

proportion  as  their  masses  are  increased,  and  as  the  square  of 
their  distance  is  decreased ;  and  it  decreases  in  proportion  as 
their  masses  are  decreased,  and  as  the  square  of  their  distance 
is  increased." 

(110.)  Having  explained  the  law  of  gravitation,  we  shall 
now  proceed  to  show  how  the  existence  of  this  force  is  prov- 
ed, and  its  law  discovered. 

The  earth  is  known  to  be  a  globular  mass  of  matter,  in- 
comparably greater  than  any  of  the  detached  bodies  which 
are  found  upon  its  surface.  If  one  of  these  bodies,  suspend- 
ed at  any  proposed  height  above  the  surface  of  the  earth,  be 
disengaged,  it  will  be  observed  to  descend  perpendicularly  to 
the  earth,  that  is,  in  the  direction  of  the  earth's  centre.  The 
force  with  which  it  descends  will  also  be  found  to  be  in  pro- 
portion to  the  mass,  without  any  regard  to  the  species  of  the 
body.  These  circumstances  are  consistent  with  the  account 
which  we  have  given  of  gravitation.  But  by  that  account  we 
should  expect,  that  as  the  falling  body  is  attracted  with  a  cer- 
tain force  towards  the  earth,  the  oarth  itself  should  be  attract- 
ed towards  it  by  the  same  force ;  and  instead  of  the  falling 
body  moving  towards  the  earth,  which  is  the  phenomenon 
observed,  the  earth  and  it  should  move  towards  each  other, 
and  meet  at  some  Liter  modi  ate  point.  This,  in  fact,  is  the 
•ase,  although  it  is  impossible  to  render  the  motion  of  the 
earth  observable,  for  reasons  which  will  easily  be  understood. 

Since  all  the  bodies  around  us  participate  in  this  motion,  it 
would  not  be  directly  observable,  even  though  its  quantity 
were  sufficiently  great  to  be  perceived  under  other  circum- 
stances. But,  setting  aside  this  consideration,  the  space 
through  which  the  earth  moves  in  such  a  case  is  too  minute  to 
be  the  subject  of  sensible  observation.  It  has  been  stated 
(107),  that  when  two  bodies  attract  each  other,  the  space, 
through  which  the  greater  approaches  the  lesser,  bears  to 
that  through  which  the  lesser  approaches  the  greater,  the 
same  proportion  as  the  mass  of  the  lesser  bears  to  the  mass 
of  the  greater.  Now  the  mass  of  the  earth  is  more  than 
1000,000,000,000,000  times  the  mass  of  any  body  which 
is  observed  to  fall  on  its  surface ;  and,  therefore,  if  even 
the  largest  body  which  can  come  under  observation,  were  to 
fall  through  an  height  of  500  feet,  the  corresponding  mo- 
tion of  the  earth  would  be  through  a  space  less  than  the 
1000,000,000,000,000th  part  of  500  feet,  which  is  less  than 
the  100,000,000,000th  part  of  an  inch. 


CHAP.    VI.  GRAVITATION.  67 

The  attraction  between  the  earth  and  detached  bodies  on 
its  surface  is  not  only  exhibited  by  the  descent  of  these 
bodies  when  unsupported,  but  by  their  pressure  when  sup- 
ported. This  pressure  is  what  is  called  weight.  The  phe- 
nomena of  weight,  and  the  descent  of  heavy  bodies,  will  be 
fully  investigated  in  the  next  chapter. 

(111.)  It  is  not  alone  by  the  direct  fall  of  bodies,  that  the 
gravitation  of  the  earth  is  manifested.  The  curvilinear 
motion  of  bodies  projected  in  directions  different  from  the 
perpendicular,  is  a  combination  of  the  effects  of  the  uniform 
velocity  which  has  been  given  to  the  projectile  by  the  impulse 
which  it  has  received,  and  the  accelerated  velocity  which  it 
receives  from  the  earth's  attraction.  Suppose  a  body  placed 
at  any  point  ¥,Jig.  21.,  above  the  surface  of  the  earth,  and 
let  P  C  be  the  direction  of  the  earth's  centre.  If  the  body 
were  allowed  to  move  without  receiving  any  impulse,  it  would 
descend  to  the  earth  in  the  direction  P  A,  with  an  acceler- 
ated motion.  But  suppose  that,  at  the  moment  of  its  depart- 
ure from  P,  it  receives  an  impulse  in  the  direction  P  B, 
whicli  would  carry  it  to  B  in  the  time  the  body  would  fall 
from  P  to  A  ;  then,  by  the  composition  of  motion,  the  body 
must,  at  the  end  of  that  time,  be  found  in  the  line  B  D, 
parallel  to  P  A.  If  the  motion  in  the  direction  of  P  A  were 
uniform,  the  body  P  would,  in  this  case,  move  in  the  straight 
line  from  P  to  D.  But  this  is  not  the  case.  The  velocity 
of  the  body  in  the  direction  P  A  is  at  first  so  small  as  to  pro- 
duce very  little  deflection  of  its  motion  from  the  line  P  B. 
As  the  velocity,  however,  increases,  this  deflection  increases, 
so  that  it  moves  from  P  to  D  in  a  curve,  which  is  convex  to- 
wards P  B. 

The  greater  the  velocity  of  the  projectile  in  the  direction 
P  B,  the  greater  sweep  the  curve  will  take.  Thus  it  will  suc- 
cessively take  the  forms  P  D,  P  E,  P  F,  &c. :  and  that  veloci- 
ty can  be  computed,  which  (setting  aside  the  resistance  of  the 
air)  would  cause  the  projectile  to  go  completely  round  the 
earth,  and  return  to  the  point  P  from  which  it  departed.  In 
thi$  case,  the  body  P  would  continue  to  revolve  round  the 
earth  like  the  moon.  Hence  it  is  obvious,  that  the  phenom- 
enon of  the  revolution  of  the  moon  round  the  earth,  is  noth- 
ing more  than  the  combined  effects  of  the  earth's  attraction, 
and  the  impulse  which  it  received  when  launched  into  space 
by  the  hand  of  its  Croator. 


Ob  THE    ELEMENTS    OF    MECHANICS.  CHAP.    VI. 

(112.)  This  is  a  great  step  in  the  analysis  of  the  phenom- 
enon of  gravitation.  We  have  thus  reduced  to  the  same 
class  two  effects  apparently  very  dissimilar — the  rectilinear 
descent  of  a  heavy  body,  and  the  nearly  circular  revolution 
of  the  moon  round  the  earth.  Hence  we  are  conducted  to 
a  generalization  still  more  extensive. 

As  the  moon's  revolution  round  the  earth,  in  an  orbit 
nearly  circular,  is  caused  by  the  combination  of  the  earth's 
attraction,  and  aa  original  projectile  impulse,  so  also  the 
similar  phenomena  of  the  planets'  revolution  round  the  sun 
in  orbits  nearly  .circular,  must  be  considered  an  effect  of  the 
same  class,  as  well  as  tihe  revolution  of  the  satellites  of  those 
planets  which  are  attended  by  such  bodies.  Although  the 
orbits  in  which  die  comets  move,  deviate  very  much  from 
circles,  yet  this  does  not  hinder  the  application  of  the  same 
principle  to  them,  their  deviation  from  circles  not  depending 
on  the  sun's  attraction^  b.i«t  only  on  the  direction  and  force 
of  the  original  impulse  which  put  them  in  motion. 

(113.)  We  therefore  conclude  that  gravitation  is  the 
principle  which,  as  it  were,  animates  the  universe.  All  the 
great  changes  and  revolutions  of  the  bodies  which  compose 
our  system,  can  be  graced  to  or  derived  from  this  principle. 
It  still  remains  to  .show  how  that  remarkable  law,  by  which 
his  force  is  declared  to  increase  or  decrease  in  the  same  pro- 
portion as  the  square  of  the  distance  from  the  attracting  body 
is  decreased  or  increased,  may  l>e  verified  and  established. 

It  has  been  shown,  that  the  curvilinear  path  of  a  projectile 
depends  on,  and  can  be  derived,  by  mathematical  reasoning, 
from  the  consideration  of  the  intensity  of  the  earth's  attrac- 
tion, and  the  force  of  the  original  impulse,  or  the  velocity  of 
projection.  In  the  same  manner,  by  a  reverse  process,  when 
we  know  the  curve  in  which  a  projectile  moves,  we  can  in- 
fer the  amount  of  the  attracting  force  which  gives  the  curva- 
ture to  its  path.  In  this  way,  from  our  knowledge  of  the 
curvature  of  the  moon's  orbit,  and  the  velocity  with  which 
she  moves,  the  intensity  of  the  attraction  which  the  earth 
exerts  upon  her  can  be  exactly  ascertained.  Upon  compar- 
ing this  with  the  force  of  gravitation  at  the  earth's  surface, 
it  is  found  that  the  latter  is  as  many  times  greater  than  the 
former,  as  the  square  of  the  moon's  distance  is  greater  than 
the  square  of  the  distance  of  a  body  on  the  surface  of  the 
earth  from  its  centre. 


CHAP.    VI.  LAW    OF    GRAVITATION. 

(114.)  If  this  were  the  only  fact  which  could  be  brought 
to  establish  the  law  of  nature,  it  might  bo  thought  to  be  an 
accidental  relation,  not  necessarily  characterizing  the  at- 
traction of  gravitation.  Upon  examining  the  orbits  and  ve- 
locities of  the  several  planets,  the  same  result  is,  however, 
obtained.  It  is  found  that  the  forces  with  which  they  are 
severally  attracted  by  the  sun  are  great,  in  exactly  the  same 
proportion  as  the  squares  of  the  several  numbers  expressing 
their  distances  are  small.  The  mutual  gravitation  of  bodies 
on  the  surface  of  the  earth  towards  each  other  is  lost  in  the 
predominating  force  exerted  by  the  earth  upon  all  of  them. 
Nevertheless,  in  some  cases,  this  effect  has  not  only  been 
observed,  but  actually  measured. 

A  plumb-line,  under  ordinary  circumstances,  hangs  in  a 
direction  truly  vertical ;  but  if  it  be  near  a  large  mass  of 
matter,  as  a  mountain,  it  has  been  observed  to  be  deflected 
from  the  true  vertical,  towards  the  mountain.  This  effect 
was  observed  by  Dr.  Maskeline  near  the  mountain  called 
Skehallien,  in  Scotland,  and  by  French  astronomers  near 
Chimbora9O.  For  particulars  of  these  observations,  see  our 
treatise  on  GEODESY. 

Cavendish  succeeded  in  exhibiting  the  effects  of  the  mutual 
gravitation  of  metallic  spheres.  Two  globes  of  lead  A,  B, 
each  about  a  foot  in  diameter,  were  placed  at  a  certain  dis- 
tance asunder.  A  light  rod,  to  the  ends  of  which  were 
attached  small  metallic  balls,  G,  D,  was  suspended  at  its 
centre  E  from  a  fine  wire,  and  the  rod  was  placed  as  in 
fig.  22.,  so  that  the  attractions  of  each  of  the  leaden  globes 
had  a  tendency  to  turn  the  rod  round  the  centre  E  in  the 
same  direction.  A  manifest  effect  was  produced  upon  the 
balls  C,  D,  by  the  gravitation  of  the  spheres.  In  this  ex- 
periment, care  must  be  taken  that  no  magnetic  substance 
is  intermixed  with  the  materials  of  the  balls. 

Having  so  far  stated  the  principles  on  which  the  law  of 
gravitation  is  established,  we  shall  dismiss  this  subject  without 
further  details,  since  it  more  properly  belongs  to  the  subject 
of  PHYSICAL  ASTRONOMY  ;  to  which  we  refer  the  reader  for 
a  complete  demonstration  of  the  law,  and  for  the  detailed 
developement  of  its  various  and  important  consequences. 


70  THE  ELEMENTS  OF  MECHANICS.      CHAP.  VII. 

CHAPTER   VII. 

TERRESTRIAL    GRAVITY 

- 

(115.)  GRAVITATION  is  the  general  name  given  to  this 
attraction,  by  whatever  masses  of  matter  it  may  be  manifested. 
As  exhibited  in  the  effects  produced  by  the  earth  upon  sur- 
rounding bodies,  it  is  called  "  terrestrial  gravity." 

As  the  attraction  of  the  earth  is  directed  towards  its  centre, 
it  might  be  expected  that  two  plumb-lines  should  appear  not 
to  be  parallel,  but  so  inclined  to  each  other  as  to  converge  to 
a  point  under  the  surface  of  the  earth.  Thus,  if  A  B  and 
C  D,Jig.  23.,  be  two  plumb-lines,  each  will  be  directed  to 
the  centre  O,  where,  if  their  directions  were  continued,  they 
would  meet.  In  ;like  manner,  if  two  bodies  were  allowed 
to  fall  from  A  and  C,  they  would  descend  in  the  directions 
A  B  and  C  D,  which  converge  to  O.  Observation,  on  the 
contrary,  shows  that  plumb-lines  suspended  in  places  not  far 
distant  from  each  other  are  truly  parallel ;  and  that  bodies  al- 
lowed to  fall,  descend  in  parallel  lines.  This  apparent  paral- 
lelism of  the  direction  of  terrestrial  gravity  is  accounted  for  by 
the  enormous  proportion  which  the  magnitude  of  the  earth 
bears  to  the  distance  between  the  two  plumb-lines  or  the  twjD 
falling  bodies  which  are  compared.  If  the  distance  betweeTT 
the  places  B,  D,  were  1200  feet,  the  inclination  of  the  lines 
A  B  and  C  D  would  not  amount,  .to  a  quarter  of  a  minute,  or 
fhe  240th  part  of  a  degree.  But  the  distance,  in  cases  where 
the  parallelism  is  assumed,  is  never  greater  than,  and  seldom 
so  great  as,  a  few  yards ;  and  hence  the  inclination  of  the 
directions  A  B  and  C  D  is  too  small  to  be  appreciated  by  any 
practical  measure.  In  the  investigation  of  the  phenomena 
of  falling  bodies,  we  shall,  therefore,  assume  that  all  the  par- 
ticles of  the  same  body  are  attracted  in  parallel  directions, 
perpendicular  to  an  horizontal  plane. 

(116.)  Since  the  intensity  of  terrestrial  gravity  increases 
as  the  square  of  the  distance  decreases,  it  might  be  expected 
that,  as  a  falling  body  approaches  the  earth,  the  force  which 
accelerates  it  should  be  continually  increasing,  and,  strictly 
speaking,  it  is  so.  But  any  height  through  which  we  observe 
falling  bodies  to  descend  bears  so  very  small  a  proportion  to 
the  whole  distance  from  the  centre,  that  the  change  of  inten- 
sity of  the  force  of  gravitv  is  quite  beyond  any  oractka! 
9 


CHAP.    VII.         BODIES    FALL    WITH    EQUAL    SPEED.  71 

means  of  estimating  it.  The  radius,  or  the  distance  from  the 
surface  of  the  earth  to  its  centre,  is  4000  miles.  Now,  sup- 
pose a  body  descended  through  the  height  of  half  a  mile,  a 
distance  very  much  beyond  those  used  in  experimental  in- 
quiries ;  the  distances  from  the  centre,  at  the  beginning  and 
end  of  the  fall,  are  then  in  the  proportion  of  8000  to  8001 ,  and 
therefore  the  proportion  of  the  force  of  attraction  at  the  com- 
mencement to  the  force  at  the  end,  being  that  of  the  squares 
of  these  numbers,  is  64,000,000  to  64,016,001,  which,  in  the 
whole  descent,  is  an  increase  of  about  one  part -in  4000;  a 
quantity  practically  insignificant.  We  shall,  therefore,  in 
explaining  the  laws  of  falling  bodies,  assume  that,  in  the 
entire  descent,  the  body  is  urged  by  a  force  of  uniform  in- 
tensity. 

Although  the  force  which  attracts  all  parts  of  the  same 
body  during  its  descent  in  a  given  place  is  the  same,  yet  the 
force  of  gravity,  at  different  parts  of  the  earth's  surface,  has 
different  intensities.  The  intensity  diminishes  with  the  lati- 
tude, so  that  it  is  greater  towards  the  poles,  and  lesser  to- 
wards the  equator.  The  causes  of  this  variation,  its  law,  and 
the  experimental  proofs  of  it,  will  be  explained  when  we 
shall  treat  of  centrifugal  force,  and  the  motion  of  pendulums 
It  is  sufficient  merely  to  advert  to  it  in  this  place. 

(117.)  Since  the  earth's  attraction  acts  separately  and 
equally  on  every  particle  of  matter,  without  regard  to  the 
nature  or  species  of  the  body,  it  follows  that  all  bodies,  of 
whatever  kind,  or  whatever  be  their  masses,  must  be  moved 
with  the  same  velocity.  If  two  equal  particles  of  matter  be 
placed  at  a  certain  distance  above  the  surface  of  the  earth, 
they  will  fall  in  parallel  lines,  and  with  exactly  the  same 
speed,  because  the  earth  attracts  them  equally.  In  the  same 
manner,  a  thousand  particles  would  fall  with,  equal  velocities 
Now,  these  circumstances  will  in  no  wise  be  changed,  if 
those  1000  particles,  instead  of  existing  separately,  be  aggre- 
gated into  two  solid  masses,  one  consisting  of  990  particles, 
and  the  other  of  10.  We  shall  thus  have  a  heavy  body  and 
a  light  one,  and,  according  to  our  reasoning,  they  must  fall 
to  the  ear tli  with  the  same  speed. 

Common  experience,  however,  is  not  always  consistent 
with  this  doctrine.  What  are  called  light  substances,  as 
feathers,  gold-leaf,  paper,  &c.,  are  observed  to  fall  slowly  and 
irregularly,  while  heavier  masses,  as  solid  pieces  of  metal, 
stones,  &,c.,  fall  rapidly.  Nay,  there  are  not  a  few  instances 


72  THE    ELEMENTS    OF    MECHANICS.  CHAP.  VII 

in  which  the  earth,  instead  of  attracting  bodies,  seems  to  re 
pel  them,  as  in  the  case  of  smoke,  vapors,  balloons,  and  other 
substances  which  actually  ascend.  We  are  to  consider  that 
the  mass  of  the  earth  is  not  the  only  agent  engaged  in  these 
phenomena.  The  earth  is  surrounded  by  an  atmosphere 
composed  of  an  elastic  or  aeriform  fluid.  This  atmosphere 
has  certain  properties,  which  will  be  explained  in  our  treatise 
on  PNEUMATICS,  and  which  are  the  causes  of  the  anomalous 
circumstances  alluded  to.  Light  bodies  rise  in  the  atmos- 
phere, for  the  same  reason  that  a  piece  of  cork  rises  from  the 
bottom  of  a  vessel  of  water ;  and  other  light  bodies  fall  more 
slowly  than  heavy  ones,  for  the  same  reason  that  an  egg  in 
water  falls  to  the  bottom  more  slowly  than  a  leaden  bullet. 
This  treatise  is  not  the  place  to  give  a  direct  explanation  of 
these  phenomena.  It  will  be  sufficient  for  our  present  pur- 
pose to  show,  that,  if  there  were  no  atmosphere,  all  bodies, 
heavy  and  light,  would  fall  at  the  same  rate.  This  may  easily 
be  accomplished  by  the  aid  of  an  air-pump.  Having,  by  that 
instrument,  abstracted  the  air  from  a  tall  glass  vessel,  we  are 
enabled,  by  means  of  a  wire  passing  air-tight  through  a  hole 
in  the  top,  to  let  fall  several  bodies  from  the  top  of  the  ves- 
sel to  the  bottom.  These,  whether  they  be  feathers,  paper, 
gold-leaf,  pieces  of  money,  &/c.,  all  descend  with  the  same 
speed,  and  strike  the  bottom  at  the  same  moment. 

(118.)  Every  one  who  has  seen  a  heavy  body  fall  from  a 
height,  has  witnessed  the  fact  that  its  velocity  increases  as  it 
approaches  the  ground.  But  if  this  were  not  observable  by 
the  eye,  it  would  be  betrayed  by  the  effects.  It  is  well 
known,  that  the  force  with  which  a  body  strikes  the  ground 
increases  with  the  height  from  whence  it  has  fallen.  This 
force,  however,  is  proportional  to  the  velocity  which  it  has  at 
the  moment  it  meets  the  ground,  and  therefore  this  velocity 
increases  with  the  height. 

When  the  observations  on  attraction  in  the  last  chapter 
are  well  understood,  it  will  be  evident  that  the  velocity  which 
a  body  has  acquired  in  falling  from  any  height,  is  the  accu- 
mulated effects  of  the  attraction  of  terrestrial  gravity  during 
the  whole  time  of  the  fall.  Each  instant  of  the  fall  a  new 
impulse  is  given  to  the  body,  from  which  it  receives  addition- 
al velocity ;  and  its  final  velocity  is  composed  of  the  aggrega- 
tion of  all  the  small  increments  of  velocity  which  are  thus 
communicated.  As  we  are  at  present  to  suppose  the  intensi- 
ty of  the  attraction  invariable,  it  will  follow  that  the  velocity 


CHAP.  VII.        DESCENT  OF  HEAVY  BODIES.  73 

communicated  to  the  body  in  each  instant  of  time  will  be 
the  same,  and  therefore  that  the  whole  quantity  of  velocity 
produced  or  accumulated  at  the  end  of  any  time  is  propor- 
tional to  the  length  of  that  time.  Thus,  if  a  certain  velocity 
be  produced  in  a  body  having  fallen  for  one  second,  twice 
that  velocity  will  be  produced  when  it  has  fallen  for  two 
seconds,  thrice  that  velocity  in  three  seconds,  and  so  on. 
Such  is  the  fundamental  principle  or  characteristic  of  uniform- 
ly accelerated  motion. 

(119.)  In  examining  the  circumstances  of  the  descent  of  a 
body,  the  time  of  the  fall,  and  the  velocity  at  each  instant  of 
that  time,  are  not  the  only  things  to  be  attended  to.  The 
spaces  through  which  it  falls  in  given  intervals  of  time,  count- 
ed either  from  the  commencement  of  its  fall,  or  from  any 
proposed  epoch  of  the  descent,  are  equally  important  objects 
of  inquiry.  To  estimate  the  space  in  reference  to  the  time 
and  the  final  velocity,  we  must  consider  that  this  space  has 
been  moved  through  with  varying  speed.  From  a  state  of 
rest  at  the  beginning  of  the  fall,  the  speed  gradually  increases 
with  the  time,  and  the  final  velocity  is  greater  still  than  that 
which  the  J,ody  had  at  any  preceding  instant  during  its  de- 
scent. We  cannot,  therefore,  directly  appreciate  the  space 
moved  through  in  this  case  by  the  time  and  final  velocity. 
But,  as  the  velocity  increases  uniformly  with  the  time,  we 
shall  obtain  the  average  speed,  by  finding  that  which  the 
body  had  in  the  middle  of  the  interval  which  elapsed  between 
the  beginning  and  end  of  the  fall,  and  thus  the  space  through 
which  the  body  has  actually  fallen  is  that  through  which  it 
#  would  move  in  the  same  time"  with  this  average  velocity  uni- 
formly continued. 

But  since  the  velocity  which  the  body  receives  in  any  time, 
counted  from  the  beginning  of  its  descent,  is  in  the  propor- 
tion of  that  time,  it  follows  that  the  velocity  of  the  body  after 
half  the  whole  time  of  descent  is  half  the  final  velocity. 
From  whence  it  appears,  that  the  height  from  which  a  body 
falls  in  any  proposed  time  is  equal  to  the  space  through  which 
a  body  would  move  in  the  same  time  with  half  the  final  ve- 
locity, and  it  is  therefore  equal  to  half  the  space  which  would 
be  moved  through  in  the  same  time  with  the  final  velocity. 

(120.)  It  follows,  from  this  reasoning,  that  between  the 
three  quantities,  the  height,  the  time,  and  the  final  velocity, 
which  enter  into  the  investigation  of  the  phenomena  of  fall- 
ing bodies,  there  are  two  fixed  relations :  First]  the  time, 
7 


74  THE  ELEMENTS  OF  MECHANICS.  CHAP.  VII 

counted  from  the  beginning  of  the  fall,  and  the  final  velocity, 
are  proportional  the  one  to  the  other ;  so  that  as  one  in- 
creases, the  other  increases  in  the  same  proportion.  Sec- 
ondly, the  height  being  equal  to  half  the  space  which  would 
be  moved  through  in  the  time  of  the  fall,  with  thejinal  veloci- 
ty, must  have  a  fixed  proportion  to  these  two  quantities,  viz. 
the  time  and  the  final  velocity,  or  must  be  proportional  to  the 
product  of  the  two  numbers  which  express  them. 

But  since  the  time  is  always  proportional  to  the  final  ve- 
locity, they  may  be  expressed  by  equal  numbers,  and  the 
product  of  equal  numbers  is  the  square  of  either  of  them. 
Hence  the  product  of  the  numbers  expressing  the  time  and 
final  velocity  is  equivalent  to  the  square  of  the  number  express- 
ing the  time,  or  to  the  square  of  the  number  expressing  the 
final  velocity.  Hence  we  infer,  that  the  height  is  always 
proportional  to  the  square  of  the  time  of  the  fall,  or  to  the 
square  of  the  final  velocity. 

(121.)  The  use  of  a  few  mathematical  characters  will  ren- 
der these  results  more  distinct,  even  to  students  not  conver- 
sant with  mathematical  science.  Let  S  express  the  height 
from  which  the  body  falls,  V  the  final  velocity,  and  T  the 
time  of  the  fall,  and  let  the  square  of  any  of  these  quantities, 
or  rather  of  their  numerical  expressions,  be  signified  by  plac- 
ing the  figure  2  over  them  ;  thus,  T2  or  V2.  The  sign  x 
between  two  numbers  signifies  that  they  are  to  be  multiplied 
together.  These  being  premised,  the  results  of  the  reason- 
ing in  which  we  have  been  just  engaged,  may  be  expressed 
as  follows : 


V  increases  proportionally  with  T 
S  -  -  -  VXT 
S T2 

s v2 


The  theorems  [3]  and  [4]  follow  from  [1]  and  [2]  ;  for 
since  by  [1]  T  is  proportional  to  V,  it  may  be  put  for  V  in 
[2],  and  by  this  substitution  V  X  T  becomes  T  X  T  or  T2. 
In  the  same  manner  and  for  the  same  reason,  V  may  be 
put  for  T,  by  which  V  X  T  becomes  V  X  V,  or  V2. 

By  these  formularies,  if  the  height  through  which  a  body 
falls  freely  in  one  second  be  known,  the  height  through 
which  it  will  fall  in  any  proposed  time  may  be  computed. 
For  since  the  height  is  proportional  to  the  square  of  the 
time  the  height  through  which  it  will  fall  in  two  seconds  will 


CHAP.  VII.        DESCENT  OF  HEAVY  BODIES.  75 

be  four  times  that  which  it  falls  through  in  one  second.  In 
three  seconds  it  will  fall  through  nine  times  that  space  ;  in 
four  seconds,  sixteen  times  ;  mjive  seconds,  twenty-Jive  times ; 
and  so  on.  The  following,  therefore,  is  a  general  rule  to 
find  the  height  through  which  a  body  will  fall  in  any  given 
time  :  "  Reduce  the  given  time  to  seconds,  take  the  square 
of  the  number  of  seconds  in  it,  and  multiply  the  height 
through  which  a  body  falls  in  one  second  by  that  number  ; 
the  result  will  be  the  height  sought." 

The  following  table  exhibits  the  heights  and  correspond- 
ing times  as  far  as  10  seconds  : 


Time     |1|2|3|4|5|6|7|8|9|10 


Height  |  1  |  4    9  |  16  |  25  |  36  |  49  |  64  |  81  |  100 


Each  unit  in  the  numbers  of  the  first  row  expresses  a  sec- 
ond of  time,  arid  each  unit  in  those  of  the  second  row  ex- 
presses the  height  through  which  a  body  falls  freely  in  a 
second. 

(122.)  If  a  body  fall  continually  for  several  successive 
seconds,  the  spaces  which  it  falls  through  in  each  succeeding 
second  have  a  remarkable  relation  among  each  other,  which 
may  be  easily  deduced  from  the  preceding  table.  Taking 
the  space  moved  through  in  the  first  second  still  as  our  unit, 
four  times  that  space  will  be  moved  through  in  the  first  two 
seconds.  Subtract  from  this  1,  the  space  moved  through 
in  the  first  second,  and  the  remainder  3  is  the  space  through 
which  the  body  falls  in  the  second  second.  In  like  manner 
if  4,  the  height  fallen  through  in  the  first  two  seconds,  be 
subtracted  from  9,  the  height  fallen  through  in  the  first 
three  seconds,  the  remainder  5  will  be  the  space  fallen 
through  in  the  third  second.  To  find  the  space  fallen  through 
in  the  fourth  second,  subtract  9,  the  space  fallen  through 
in  the  first  three  seconds,  from  16,  the  space  fallen  through 
in  the  first  four  seconds,  and  the  result  is  7,  and  so  on. 

It  thus  appears  that  if  the  space  fallen  through  in  the  first 
second  be  called  1,  the  spaces  described  in  the  second,  third, 
fourth,  fifth,  &/c.  seconds,  will  be  expressed  by  the  odd  num- 
bers respectively,  3,  5,  7,  9,  &c.  This  places  in  a  striking 
point  of  view  the  accelerated  motion  of  a  falling  body,  the 
spaces  moved  through  in  each  succeeding  second  being  con- 
tinually increased. 

(123.)  If  velocity  be  estimated  by  the  space  through  which 


76  THE  ELEMENTS  OF  MECHANICS.  CHAP.  VII. 

the  body  would  move  uniformly  in  one  second,  then  the  final 
velocity  of  a  body  falling  for  one  second  will  be  2  ;  for  with 
that  final  velocity  the  body  would  in  one  second  move  through 
twice  the  height  through  which  it  has  fallen. 

(124.)  Since  the  final  velocity  increases  in  the  same  pro- 
portion as  the  time,  it  follows  that  after  two  seconds  it  is 
twice  its  amount  after  one,  and  after  three  seconds  thrice 
that,  and  so  on.  Thus  the  following  table  exhibits  the  final 
velocities  corresponding  to  the  times  of  descent : 


Time  I  1  I  2  |  3    4|5|6|7|8|9 


Final  velocity  |  2  |  4  |  6  |  8  |  10  |  12    14  |  16  |  18  |  20 


The  numbers  in  the  second  row  express  the  spaces  through 
which  a  body  with  the  final  velocity  would  move  in  one 
second,  the  unit  being,  as  usual,  the  space  through  which  a 
body  falls  freely  in  one  second. 

(125.)  Having  thus  developed  theoretically  the  laws  which 
characterize  the  descent  of  bodies,  falling  freely  by  the  force 
of  gravity,  or  by  any  other  uniform  force  of  the  same  kind, 
it  is  necessary  that  we  should  show  how  these  laws  can  be 
exhibited  by  actual  experiment.  There  are  some  circum- 
stances attending  the  fall  of  heavy  bodies  which  would  ren- 
der it  difficult,  if  not  impossible,  to  illustrate,  by  the  direct 
observation  of  this  phenomenon,  the  properties  which  have 
been  explained  in  this  chapter.  A  body  falling  freely  by 
the  force  of  gravity,  as  we  shall  hereafter  prove,  descends 
in  one  second  of  time  through  a  height  of  about  16  feet  ; 
in  two  seconds,  it  would,  therefore,  fall  through  four  times 
that  space,  or  64  feet ;  in  three  seconds,  through  9  times 
the  height,  or  144  feet ;  and  in  four  seconds,  through  256 
feet.  In  order,  therefore,  to  be  enabled  to  observe  the  phe- 
nome.na  for  only  four  seconds,  we  should  command  an  height 
of  at  least  256  feet.  But  further ;  the  velocity  at  the  end 
of  the  first  second  would  be  at  the  rate  of  32  feet  per  second ; 
at  the  end  of  the  second  second,  it  would  be  64  feet  per 
second  ;  and  towards  the  end  of  the  fall,  it  would  be  about 
120  feet  per  second.  It  is  evident  that  this  great  degree  of 
rapidity  would  be  a  serious  impediment  to  accurate  observa- 
tion, even  though  we  should  be  able  to  command  the  requi- 
site height. 

It  occurred  to  Mr.  George  Attwood,  a  mathematician  and 
natural  philosopher  of  the  last  century,  that  all  the  phenomena 


CHAP.  vii.  ATTWOOD'S  MACHINE.  77 

of  falling  bodies  might  be  experimentally  exhibited  and  ac- 
curately observed,  if  a  force  of  the  same  kind  as  gravity,  viz. 
an  uniformly  accelerating  force,  be  used,  but  of  a  much  less 
intensity ;  so  that,  while  the  motion  continues  to  be  governed 
by  the  same  laws,  its  quantity  may  be  so  much  diminished, 
that  the  final  velocity,  even  after  a  descent  of  many  seconds, 
shall  be  so  moderated  as  to  admit  of  most  deliberate  and 
exact  observation.  This  being  once  accomplished,  nothing 
more  would  remain  but  to  find  the  height  through  which 
a  body  would  fall  in  one  second,  or,  what  is  the  same,  the 
proportion  of  the  force  of  gravity  to  the  mitigated  but  uni- 
formly by  accelerating  force  thus  substituted  for  it. 

(136:)  To  realize  this  notion,  Attwood  constructed  a 
wheel  turning  on  its  axle  with  very  little  friction,  and  hav- 
ing a  groove  on  its  edge  to  receive  a  string.  Over  this 
wheel,  and  in  the  groove,  he  placed  a  fine  silken  cord,  to 
the  ends  of  which  were  attached  equal  cylindrical  weights. 
Thus  placed,  the  weights  perfectly  balance  each  other,  and 
no  motion  ensues.  To  one  of  the  weights  he  then  added 
a  small  quantity,  so  as  to  give  it  a  slight  preponderance. 
The  loaded  weight  now  began  to  descend,  drawing  up  on 
the  other  side  the  unloaded  weight.  The  descent  of  the 
loaded  weight,  under  these  circumstances,  is  a  motion  ex- 
actly of  the  same  kind  as  the  descent  of  a  heavy  body 
falling  freely  by  the  force  of  gravity ;  that  is,  it  increases 
according  to  the  same  laws,  though  at  a  very  diminished 
rate.  To  explain  this,  suppose  that  the  loaded  weight  de- 
scends from  a  state  of  rest  through  one  inch  in  a  second ; 
it  will  descend  through  4  inches  in  two  seconds,  through  9  in 
three,  through  16  in  four,  and  so  on.  Thus,  in  20  seconds, 
it  would  descend  through  400  inches,  or  33  feet  4  inches — 
a  height  which,  if  it  were  necessary,  could  easily  be  com- 
manded. 

It  might,  perhaps,  be  thought,  that,  since  the  weights  sus- 
pended at  the  ends  of  the  thread  are  in  equilibrium,  and 
therefore  have  no  tendency  either  to  move  or  to  resist  mo- 
tion, the  additional  weight  placed  upon  one  of  them  ought  to 
descend  as  rapidly  as  it  would  if  it  were  allowed  to  fall  freely 
and  unconnected  with  them.  It  is  very  true  that  this  weight 
will  receive  from  the  attraction  of  the  earth  the  same  force 
when  placed  upon  one  of  the  suspended  weights,  as  it  would 
if  it  were  disengaged  from  them  ;  but  in  the  consequences 
'which  ensue,  there  is  this  difference.  If  it  were  unconncct- 
7* 


78  THE  ELEMENTS  OF  MECHANICS.  CHAP.  VII. 

ed  with  the  suspended  weights,  the  whole  force  impressed 
upon  it  would  be  expended  in  accelerating  its  descent ;  but, 
being  connected  with  the  equal  weights  which  sustain  each 
other  in  equilibrium,  by  the  silken  cord  passing  over  the 
wheel,  the  force  which  is  impressed  upon  the  added  weight 
is  expended,  not,  as  before,  in  giving  velocity  to  the  added 
weight  alone,  but  to  it  together  with  the  two  equal  weights 
appended  to  the  string,  one  of  which  descends  with  the  added 
weight,  and  the  other  rises  on  the  opposite  side  of  the  wheel. 
Hence,  setting  aside  any  effect  which  the  wheel  itself  pro- 
duces, the  velocity  of  the  descent  must  be  lessened  just  in 
proportion  as  the  mass  among  which  the  impressed  force  is 
to  be  distributed  is  increased  ;  and  therefore  the  rate  of  the 
fall  bears  to  that  of  a  body  falling  freely  the  same  proportion 
as  the  added  weight  bears  to  the  sum  of  the  masses  of  the 
equal  suspended  weights  and  the  added  weight.  Thus  the 
smaller  the  added  weight  is,  and  the  greater  the  equal  sus- 
pended weights  are,  tire  slower  will  the  rate  of  descent  be. 

To  render  the  circumstances  of  the  fall  conveniently  ob- 
servable, a  vertical  shaft  (see  ^'^.24.)  is  usually  provided, 
which  is  placed  behind  the  descending  weight.  This  pillar 
is  divided  into  inches  and  halves,  and,  of  course,  may  be  still 
more  minutely  graduated,  if  necessary.  A  stage  to  receive 
the  falling  weight  is  movable  on  this  pillar,  and  capable  of 
being  fixed  in  any  proposed  position  by  an  adjusting  screw. 
A  pendulum  vibrating  seconds,  the  beat  of  which  ought  to  be 
very  audible,  is  placed  near  the  observer.  The  loaded  weight 
being  thus  allowed  to  descend  for  any  proposed  time,  or  from 
any  required  height,  all  the  circumstances  of  the  descent 
may  be  accurately  observed,  and  the  several  laws  already 
explained  in  this  chapter  may  be  experimentally  verified. 

(127.)  The  laws  which  govern  the  descent  of  bodies  by 
gravity,  being  reversed,  will  be  applicable  to  the  ascent  of 
bodies  projected  upwards.  If  a  body  be  projected  directly 
upwards  with  any  given  velocity,  it  will  rise  to  the  height 
from  which  it  should  have  fallen  to  acquire  that  velocity.  The 
earth's  attraction  will,  in  this  case,  gradually  deprive  the 
body  of  the  velocity  which  is  communicated  to  it  at  the  mo- 
ment at  which  it  is  projected.  Consequently,  the  phenome- 
non will  be  that  of  retarded  motion.  At  each  part  of  its 
ascent,  it  will  have  the  same  velocity  which  it  would  have 
if  it  descended  to  the  same  place  from  the  highest  point  to 
which  it  rises.  Hence  it  is  clear,  that  all  the  particulars 


CHAP.  VIII.  MOTION  ON  INCLINED  PLANES.  79 

relative  to  the  ascent  of  bodies  may  be  immediately  inferred 
from  those  of  their  descent,  and  therefore  this  subject  de- 
mands no  further  notice. 

To  complete  the  investigation  of  the  phenomena  of  falling 
bodies,  it  would  now  only  remain  to  explain  the  method  of 
ascertaining  the  exact  height  through  which  a  body  would 
descend  in  one  second,  if  unresisted  by  the  atmosphere,  or 
any  other  disturbing  cause.  As  the  solution  of  this  problem, 
however,  requires  the  aid  of  principles  not  yet  explained,  it 
must  for  the  present  be  postponed. 


CHAPTER  VIII. 

OF  THE  MOTION  OF  BODIES  ON  INCLINED  PLANES  AND  CURVES. 

(128.)  IN  the  last  chapter,  we  investigated  the  phenomena 
of  bodies  descending  freely  in  the  vertical  direction,  and  de- 
termined the  laws  which  govern,  not  their  motion  alone,  but 
that  of  bodies  urged  by  any  uniformly  accelerating  force 
whatever.  We  shall  now  consider  some  of  the  most  ordinary 
cases  in  which  the  free  descent  of  bodies  is  impeded,  and 
the  effects  of  their  gravitation  modified. 

(129.)  If  a  body,  urged  by  any  forces  whatever,  be  placed 
upon  a  hard,  unyielding  surface,  it  will  evidently  remain  at 
rest,  if  the  resultant  (76)  of  all  the  forces  which  are  applied 
to  it  be  directed  perpendicularly  against  the  surface.  In 
this  case,  the  effect  produced  is  pressure,  but  no  motion  en- 
'ues.  If  only  one  force  act  upon  the  body,  it  will  remain  a* 
rest,  provided  the  direction  of  that  force  be  perpendicular  to 
the  surface. 

But  the  effect  will  be  different,  if  the  resultant  of  the 
forces  which  are  applied  to  the  body  be  oblique  to  the  sur- 
face. In  that  case,  this  resultant,  which,  for  simplicity,  may 
be  taken  as  a  single  force,  may  be  considered  as  mechanically 
equivalent  to  two  forces  (76),  one  in  the  direction  of  the 
surface,  and  the  other  perpendicular  to  it.  The  latter  ele- 
ment will  be  resisted,  and  will  produce  a  pressure ;  the  former 
will  cause  the  body  to  move.  This  will  perhaps  be  more 
clearly  apprehended  by  the  aid  of  a  diagram. 

Let  A  B,  Jig.  25.,  be  the  surface,  and  let  P  be  a  particle 
of  matter  placed  upon  it,  and  urged  by  a  force  in  the  direo- 


80  THE    ELEMENTS    OP    MECHANICS.  CHAP.  VIII. 

tion  P  D,  perpendicular  to  A  B.  It  is  manifest,  that  this 
force  can  only  press  the  particle  P  against  A  B,  but  cannot 
give  it  any  motion. 

But  let  us  suppose,  that  the  force  which  urges  P  is  in  a 
direction  P  F,  oblique  to  A  B.  Taking  P  F  as  the  diagonal 
of  a  parallelogram,  whose  sides  are  P  D  and  P  C  (74),  the  force 
P  F  is  mechanically  equivalent  to  two  forces,  expressed  by 
the  lines  P  D  and  P  C.  But  P  D,  being  perpendicular  to  A  B, 
produces  pressure  without  motion,  and  P  C,  being  in  the 
direction  of  A  B,  produces  motion  without  pressure.  Thus 
the  effect  of  the  force  P  F  is  distributed  between  motion  and 
pressure  in  a  certain  proportion,  which  depends  on  the  ol>- 
liquity  of  its  direction  to  that  of  the  surface.  The  two  ex- 
treme cases  are,  1.  When  it  is  in  the  direction  of  the  surface  ; 
it  then  produces  motion  without  pressure  :  and,  2.  When 
it  is  perpendicular  to  the  surface  ;  it  then  produces  pressure 
without  motion.  In  all  intermediate  directions,  however,  it 
will  produce  both  these  effects. 

(130.)  It  will  be  very  apparent,  that  the  more  oblique  the 
direction  of  the  force  P  F  is  to  A  B,  the  greater  will  be  that 
part  of  it  which  produces  motion,  and  the  less  will  that  be 
which  produces  pressure.  This  will  be  evident  by  inspecting 
Jig.  26.  In  this  figure,  the  line  P  F,  which  represents  the 
force,  is  equal  to  P  F  in  Jig.  25.  But  P  D,  which  expresses 
the  pressure,  is  less  in  Jig.  20.  than  in  Jig.  25.,  while  P  C, 
which  expresses  the  motion,  is  greater.  So  long,  then,  a.s 
the  obliquity  of  the  directions  of  the  surface  and  the  force 
remain  unchanged,  so  long  will  the  distribution  of  the  force 
between  motion  and  pressure  remain  the  same  ;  and  there- 
fore, if  the  force  itself  remain  the  same,  the  parts  of  it  which 
produce  motion  and  pressure  will  be  respectively  equal. 

(131.)  These  general  principles  being  understood,  no  dif- 
ficulty can  arise  in  applying  them  to  the  motion  of  bodies 
urged  on  inclined  planes  or  curves  by  the  force  of  gravity. 
If  a  body  be  placed  on  an  unyielding  horizontal  plane,  it 
will  remain  at  rest,  producing  a  pressure  on  the  plane  equal 
to  the  total  amount  of  its  weight.  For,  in  this  case,  the  force 
which  urges  the  body  being  that  of  terrestrial  gravity,  its 
direction  is  vertical,  and  therefore  perpendicular  to  the  hori- 
zontal plane. 

But  if  the  body  P,  Jig.  25.,  be  placed  upon  a  plane  A  B, 
oblique  to  the  direction  of  the  force  of  gravity,  then,  accord- 
ing to  what  has  been  proved  (129),  the  weight  of  the  body 


CHAP.  VIII.  MOTION  ON  INCLINED  PLANES.  81 

will  be  distributed  into  two  parts,  P  C  and  P  D ;  one,  P  D,  pro- 
ducing a  pressure  on  the  plane  A  B,  and  the  other,  P  C,  produc- 
ing motion  down  the  plane.  Since  the  obliquity  of  the  perpen- 
dicular direction  P  F  of  the  weight  to  that  of  the  plane  A  B 
must  be  the  same  on  whatever  part  of  the  plane  the  weight 
may  be  placed,  it  follows  (130),  that  the  proportion  P  C  of 
the  weight  which  urges  the  body  down  the  plane,  must  be 
the  same  throughout  its  whole  descent. 

(132.)  Hence  it  may  easily  be  inferred,  that  the  force 
down  the  plane  is  uniform  ;  for  since  the  weight  of  the  body 
P  is  always  the  same,  and  since  its  proportion  to  that  part 
which  urges  it  down  the  plane  is  the  same,  it  follows  that 
the  quantity  of  this  part  cannot  vary.  The  motion  of  a  heavy 
body  down  an  inclined  plane  is  therefore  an  uniformly  accel- 
erated motion,  and  is  characterized  by  all  the  properties  of 
uniformly  accelerated  motion,  explained  in  the  last  chapter. 

Since  P  F  represents  the  force  of  gravity,  that  is,  the  force 
with  which  the  body  would  descend  freely  in  the  vertical 
direction,  and  P  C  the  force  with  which  it  moves  down  the 
plane,  it  follows  that  a  body  would  fall  freely  in  the  vertical 
direction  from  P  to  F  in  the  same  time  as  on  the  plane  it 
would  move  from  P  to  C.  In  this  manner,  therefore,  when 
the  height  through  which  a  body  would  fall  vertically  is  known, 
the  space  through  which  it  would  descend  in  the  same  time 
down  any  given  inclined  plane  may  be  immediately  deter- 
mined. For  let  A  B,  Jig.  25.,  be  the  given  inclined  plane, 
and  let  P  F  be  the  space  through  which  the  body  would  fall 
in  one  second.  From  F  draw  F  C  perpendicular  to  the  plane, 
and  the  space  P  C  is  that  through  which  the  body  P  will  fall 
in  one  second  on  the  plane. 

(133.)  As  the  angle  BAH,  which  measures  the  elevation 
of  the  plane,  is  increased,  the  obliquity  of  the  vertical  direc- 
tion P  F  with  the  plane  is  also  increased.  Consequently, 
according  to  what  has  been  proved  (130),  it  follows,  that,  as 
the  elevation  of  the  plane  is  increased,  the  force  which  urges 
the  body  down  tho  plane  is  also  increased,  and  as  the  eleva- 
tion is  diminished,  the  force  suffers  a  corresponding  diminu- 
tion. The  two  extreme  cases  are,  1.  When  the  plane  is 
raised  until  it  becomes  perpendicular,  in  which  case  the 
weight  is  permitted  to  fall  freely,  without  exerting  any  pres- 
sure upon  the  plane ;  and,  2.  When  the  plane  is  depressed 
until  it  becomes  horizontal,  in  which  case  the  whole  weight 
is  supported,  and  there  is  no  motion. 


__  I 

82  THE  ELEMENTS  OF  MECHANICS.  CHAP.  VIII. 

From  these  circumstances  it  follows,  that,  by  means  of  an 
inclined  plane,  we  can  obtain  an  uniformly  accelerating  force 
of  any  magnitude  less  than  that  of  gravity. 

We  have  here  omitted,  and  shall  for  the  present  in  every 
instance  omit,  the  effects  of  friction,  by  which  the  motion 
down  the  plane  is  retarded.  Having  first  investigated  the 
mechanical  properties  of  bodies  supposed  to  be  free  from 
friction,  we  shall  consider  friction  separately,  and  show  how 
the  present  results  are  modified  by  it. 

(134.)  The  accelerating  forces  on  different  inclined  planes 
may  be  compared  by  the  principle  explained  in  (131).  Let 
Jigs.  25.  and  26.  be  two  inclined  planes,  and  take  the  lines 
P  F  in  each  figure  equal,  both  expressing  the  force  of  gravity, 
then  P  C  will  be  the  force  which  in  each  case  urges  the  body 
down  the  plane. 

As  the  force  down  an  inclined  plane  is  less  than  that 
which  urges  a  body  falling  freely  in  the  vertical  direction, 
the  space  through  which  the  body  must  fall  to  attain  a  certain 
final  velocity  must  be  just  so  much  greater  as  the  acceler- 
ating force  is  less.  On  this  principle  we  shall  be  able  to  de- 
termine the  final  velocity  in  descending  through  any  space 
on  a  plane,  compared  with  the  final  velocity  attained  in  fall- 
ing freely  in  the  vertical  direction.  Suppose  the  body  P, 
Jig.  27.,  placed  at  the  top  of  the  plane,  and  from  H  draw 
the  perpendicular  H  C.  If  B  H  represent  the  force  of  grav- 
ity, B  C  will  represent  the  force  down  the  plane  (131).  In 
order  that  the  body  moving  down  the  plane  shall  have  a  final 
velocity  equal  to  that  of  one  which  has  fallen  freely  from 
B  to  H,  it  will  be  necessary  that  it  should  move  from  B  down 
the  plane,  through  a  space  which  bears  the  same  proportion 
to  B  H  as  B  H  does  to  B  C.  But  since  the  triangle  A  B  H 
is  in  all  respects  similar  to  H  B  C,  only  made  upon  a  larger 
scale,  the  line  A  B  bears  the  same  proportion  to  B  H  as  B  H 
bears  to  B  C.  Hence,  in  falling  on  the  inclined  plane  from 
B  to  A,  the  final  velocity  is  the  same  as  in  falling  freely  from 
BtoH. 

It  is  evident  that  the  same  will  be  true  at  whatever  level 
an  horizontal  line  be  drawn.  Thus,  if  I  K  be  horizontal, 
the  final  velocity  in  falling  on  the  plane  from  B  to  I  will 
be  the  same  as  the  final  velocity  in  falling  freely  from  B 
toK. 

(135.)  The  motion  of  a  heavy  body  down  a  curve  differs  in 
an  important  respect  from  the  motion  down  an  inclined  plane. 


i 

t 


CHAP.  VIII.  CENTRIFUGAL    FORCE.  83 

Every  part  of  the  plane  being  equally  inclined  to  the  verti- 
cal direction,  the  effect  of  gravity  in  the  direction  of  the 
plane  is  uniform  ;  and,  consequently,  the  phenomena  obey  all 
the  established  laws  of  uniformly  accelerated  motion.  If, 
however,  we  suppose  the  line  B  A,  on  which  the  body  P 
descends,  to  be  curved,  as  in  Jig.  28,  the  obliquity  of  its  di- 
rection at  different  parts,  to  the  direction  P  F  of  gravity, 
will  evidently  vary.  In  the  present  instance,  this  obliquity 
is  greater  towards  B  and  less  towards  A,  and  hence  the 
part  of  the  force  of  gravity  which  gives  motion  to  the  body  is 
greater  towards  B  than  towards  A  (130).  The  force,  there- 
fore, which  urges  the  body,  instead  of  being  uniform,  as  in 
the  inclined  plane,  is  here  gradually  diminished.  The  rate 
of  this  diminution  depends  entirely  on  the  nature  of  the  curve, 
and  can  be  deduced  from  the  properties  of  the  curve  by 
mathematical  reasoning.  The  details  of  such  an  investiga- 
tion are  not,  however,  of  a  sufficiently  elementary  character 
to  allow  of  being  introduced  with  advantage  into  this  treatise. 
We  must  therefore  limit  ourselves  to  explain  such  of  the  re- 
sults as  may  be  necessary  for  the  developement  of  the  other 
parts  of  the  science. 

(136.)  When  a  heavy  body  is  moved  down  an  inclined 
plane  by  the  force  of  gravity,  the  plane  has  been  proved  to 
sustain  a  pressure,  arising  from  a  certain  part  of  the  weight 
P  T),Jig.  25.,  which  acts  perpendicularly  to  the  plane.  This 
is  also  the  case  in  moving  down  a  curve  such  as  B  A,  Jig.  28. 
In  this  case,  also,  the  whole  weight  is  distributed  between  that 
part  which  is  directed  down  the  curve,  and  that  which,  being 
perpendicular  to  the  curve,  produces  a  pressure  upon  it. 
There  is,  however,  another  cause  which  produces  a  pressure 
upon  the  curve,  and  which  has  no  operation  in  the  case  of 
the  inclined  plane.  By  the  property  of  inertia,  when  a  body 
is  put  in  motion  in  any  direction,  it  must  persevere  in  that 
direction,  unless  it  be  deflected  from  it  by  an  efficient  force. 
In  the  motion  down  an  inclined  plane,  the  direction  is  never 
changed,  and  therefore,  by  its  inertia,  the  falling  body  retains 
all  the  motion  impressed  upon  it  continually  in  the  same 
direction ;  but  when  it  descends  upon  a  curve,  its  direction 
is  constantly  varying,  and  the  resistance  of  the  curve  being 
the  deflecting  cause,  the  curve  must  sustain  a  pressure  equal 
to  that  force  which  would  thus  be  capable  of  continually  de- 
flecting the  body  from  the  rectilinear  path  in  which  it  would 
move  in  virtue  of  its  inertia.  This  pressure  entirely  depends 


84  THE    ELEMENTS    OF    MECHANICS.  CHAP.    VIII 

on  the  curvature  of  the  path  in  which  the  body  is  constrain- 
ed to  move,  and  on  its  inertia,  and  is  therefore  altogether  in- 
dependent of  the  weight,  and  would,  in  fact,  exist  if  the  weight 
were  without  effect. 

(137.)  This  pressure  has  been  denominated  centrifugal 
force,  because  it  evinces  a  tendency  of  the  moving  body  to 
fly  from  the  centre  of  the  curve  in  which  it  is  moved.  Its 
quantity  depends  conjointly  on  the  velocity  of  the  motion  and 
the  curvature  of  the  path  through  which  the  body  is  moved. 
As  circles  may  be  described  with  every  degree  of  curvature, 
according  to  the  length  of  the  radius,  or  the  distance  from 
their  circumference  to  their  centre,  it  follows  that,  whatever 
be  the  curve  in  which  the  body  moves,  a  circle  can  always  be 
assigned  which  has  the  same  curvature  as  is  found  at  any 
proposed  point  of  the  given  curve.  Such  a  circle  is  called 
"the  circle  of  curvature"  at  that  point  of  the  curve  ;  and  as 
all  curves,  except  the  circle,  vary  their  degrees  of  curvature 
at  different  points,  it  follows  that  different  parts  of  the  same 
curve  will  have  different  circles  of  curvature.  It  is  evident 
that  the  greater  the  radius  of  a  circle  is,  the  less  is  its  curvature : 
thus  the  circle  with  the  radius  A  B,  Jig.  29.,  is  more  curved 
than  that  whose  radius  is  C  D,  and  that  in  the  exact  proportion 
of  the  radius  C  D  to  the  radius  A  B.  The  radius  of  the  circle 
of  curvature  for  any  part  of  a  curve  is  called  "  the  radius  of 
curvature"  of  that  part. 

(13^.)  The  centrifugal  pressure  increases  as  the  radius  of 
curvature  increases;  but  it  also  has  a  dependence  on  the 
velocity  with  which  the  moving  body  swings  round  the  centre 
of  the  circle  of  curvature.  This  velocity  is  estimated  either 
by  the  actual  space  through  which  the  body  moves,  or  by  the 
angular  velocity  of  a  line  drawn  from  the  centre  of  the  circle 
to  the  moving  body.  That  body  carries  one  end  of  this 
line  with  it,  while  the  other  remains  fixed  at  the  centre.  As 
this  angular  swing  round  the  centre  increases,  the  centrifugal 
pressure  increases.  To  estimate  the  rate  at  which  this  pres- 
sure in  general  varies,  it  is  necessary  to  multiply  the  square 
of  the  number  expressing  the  angular  velocity  by  that  which 
expresses  the  radius  of  curvature,  and  the  force  increases  in 
the  same  proportion  as  the  product  thus  obtained. 

(139.)  We  have  observed  that  the  same  causes  which  pro- 
duce pressure  on  a  body  restrained,  will  produce  motion  if  the 
body  be  free.  Accordingly,  if  a  body  be  moved  by  any  efficient 
cause  in  a  curve,  it  will,  by  reason  of  the  centrifugal  force, 


CHAP.   VIII.  CENTRIFUGAL    FORCE.  85 

fly  off,  and  the  moving  force  with  which  it  will  thus  retreat 
from  the  centre  round  which  it  is  whirled,  will  be  a  measure 
of  the  centrifugal  force.  Upon  this  principle  an  apparatus 
called  a  whirling  table  has  been  constructed,  for  the  purpose 
of  exhibiting  experimental  illustrations  of  the  laws  of  centrif- 
ugal force.  By  this  machine  we  are  enabled  to  place  any 
proposed  weights  at  any  given  distances  from  centres  round 
which  they  are  whirled,  either  with  the  same  angular  velocity, 
or  with  velocities  having  a  certain  proportion.  Threads  at- 
tached to  the  whirling  weights  are  carried  to  the  centres  round 
which  they  respectively  revolve,  and  there,  passing  over  pul- 
leys, are  connected  with  weights  which  may  be  varied  at 
pleasure.  When  the  whirling  weights  ily  from  their  respec- 
tive centres,  by  reason  of  the  centrifugal  force,  they  draw  up 
the  weights  attached  to  the  other  ends  of  the  threads,  and  the 
amount  of  the  centrifugal  force  is  estimated  by  the  weight 
which  it  is  capable  of  raising. 

With  this  instrument  the  following  experiments  may  be 
exhibited : — 

Exp.  1.  Equal  weights  whirled  with  the  same  velocity  at 
equal  distances  from  the  centre  raise  the  same  weight,  and 
therefore  have  the  same  centrifugal  force. 

Exp.  2.  Equal  weights  whirled  with  the  same  angular 
velocity  at  distances  from  the  centre  in  the  proportion  of  one 
to  two,  will  raise  weights  in  the  same  proportion.  Therefore 
the  centrifugal  forces  are  in  that  proportion. 

Exp.  3.  Equal  weights  whirled  at  equal  distances  with 
angular  velocities  which  are  as  one  to  two,  will  raise  weights 
as  one  to  four  ;  that  is,  as  the  squares  of  the  angular  velocities. 
Therefore  the  centrifugal  forces  are  in  that  proportion. 

Exp.  4.  Equal  weights  whirled  at  distances  which  are  as 
two  to  three,  with  angular  velocities  which  are  as  one  to  two, 
will  raise  weights  which  are  as  two  to  twelve  ;  that  is,  as  the 
products  of  the  distances  two  and  three,  and  the  squares,  one 
and  four,  of  the  angular  velocities.  Hence  the  centrifugal 
forces  are  in  this  proportion. 

The  centrifugal  force  must  also  increase  as  the  mass  of  the 
body  moved  increases  ;  for,  like  attraction,  each  particle  of  the 
moving  body  is  separately  and  equally  affected  by  it.  Hence 
a  double  mass,  moving  at  the  same  distance,  and  with  the 
same  velocity,  will  have  a  double  force.  The  following  ex- 
periment verifies  this : — 

Exp.  5.  If  weights,  which  are  as  one  to  two,  be  whirled  at 
8 


86  THE    ELEMENTS    OF    MECHANICS.  CHAP.  VIII. 

equal  distances  with  the  same  velocity,  they  will  raise  weights 
which  are  as  one  to  two. 

(140.)  The  consideration  of  centrifugal  force  proves  that 
if  a  body  be  observed  to  move  in  a  curvilinear  path,  some 
efficient  cause  must  exist  which  prevents  it  from  flying  off, 
and  which  compels  it  to  revolve  round  the  centre.  If  the  body 
be  connected  with  the  centre  by  a  thread,  cord,  or  rod,  then 
the  effect  of  the  centrifugal  force  is  to  give  tension  to  the 
thread,  cord,  or  rod.  If  an  unyielding  curved  surface  be 
placed  on  the  convex  side  of  the  path,  then  the  force  will 
produce  pressure  on  this  surface.  But  if  a  body  is  observed 
to  move  in  a  curve  without  any  visible  material  connection 
with  its  centre,  and  without  any  obstruction  on  the  convex 
side  of  its  path  to  resist  its  retreat,  as  is  the  case  with  the 
motions  of  the  planets  round  the  sun,  and  the  satellites  round 
the  planets,  it  is  usual  to  assign  the  cause  to  the  attraction  of 
the  body  which  occupies  the  centre  :  in  the  present  instance, 
the  sun  is  that  body,  and  it  is  customary  to  say  that  the  at- 
traction of  the  sun,  neutralizing  the  effects  of  the  centrifugal 
force  of  the  planets,  retains  them  in  their  orbits.  We  have 
elsewhere  animadverted  on  the  inaccurate  and  unphilosophi- 
cal  style  of  this  phraseology,  in  which  terms  are  admitted 
which  intimate  not  only  an  unknown  cause,  but  assign  its 
seat,  and  intimate  something  of  its  nature.  All  that  we 
are  entitled  to  declare  in  this  case  is,  that  a  motion  is  con- 
tinually impressed  upon  the  planet ;  that  this  motion  is  direct- 
ed towards  the  sun  ;  that  it  counteracts  the  centrifugal  force  ; 
but  from  whence  this  motion  proceeds,  whether  it  be  a  virtue 
resident  in  the  sun,  or  a  property  of  the  medium  or  space 
in  which  both  sun  and  planets  are  placed,  or  whatever  other 
influence  may  be  its  proximate  cause,  we  are  altogether  ig- 
norant. 

(141.)  Numerous  examples  of  the  effects  of  centrifugal 
force  may  be  produced. 

If  a  stone  or  other  weight  be  placed  in  a  sling,  which  is 
whirled  round  by  the  hand  in  a  direction  perpendicular  to  the 
ground,  the  stone  will  not  fall  out  of  the  sling,  even  when  it 
is  at  the  top  of  its  circuit,  and,  consequently,  has  no  support 
beneath  it.  The  centrifugal  force,  in  this  case,  acting  from 
the  hand,  which  is  the  centre  of  rotation,  is  greater  than  the 
weight  of  the  body,  and  therefore  prevents  its  fall. 

In  like  manner,  a  glass  of  water  may  be  whirled  so  rapidly, 
that,  even  when  the  mouth  of  the  glass  is  presented  down- 


CHAP.  VIII.  FAMILIAR    EXAMPLES.  87 

wards,  the  water  will  still  be  retained  in  it  by  the  centrifugal 
force. 

If  a  bucket  of  water  be  suspended  by  a  number  of  threads, 
and  these  threads  be  twisted  by  turning  round  the  bucket 
many  times  in  the  same  direction,  on  allowing  the  cords  to 
untwist,  the  bucket  will  be  whirled  rapidly  round,  and  the 
water  will  be  observed  to  rise  on  its  sides  and  sink  at  its 
centre,  owing  to  the  centrifugal  force  with  which  it  is  driven 
from  the  centre.  This  effect  might  be  carried  so  far,  that 
all  the  water  would  flow  over,  and  leave  the  bucket  nearly 
empty. 

(142.)  A  carriage,  or  horseman,  or  pedestrian,  passing  a 
corner,  moves  in  a  curve,  and  suffers  a  centrifugal  force,  which 
increases  with  the  velocity,  and  v/hich  impresses  on  the  body 
a  force  directed  from  the  corner.  An  animal  causes  its  weight 
to  resist  this  force,  by  voluntarily  inclining  its  body  towards 
the  corner.  In  this  case,  let  A  B,Jig.  30.,  be  the  body  ;  C  D 
is  the  direction  of  the  weight  perpendicular  to  the  ground, 
and  C  F  is  the  direction  of  the  centrifugal -force  parallel  to 
the  ground  and  from  the  corner.  The  body  A  B  is  inclined 
to  the  corner,  so  that  the  diagonal  force  (74),  which  is  me- 
chanically equivalent  to  the  weight  and  centrifugal  force,  shall 
be  in  tie  direction  C  A,  and  shall  therefore  produce  the  pres- 
sure of  the  feet  upon  the  ground. 

As  the  velocity  is  increased,  the  centrifugal  force  is  also 
increased,  and  therefore  a  greater  inclination  of  the  body  is 
necessary  to  resist  it.  We  accordingly  find  that  the  more 
rapidly  a  corner  is  turned,  the  more  the  animal  inclines  his 
body  towards  it. 

A  carriage,  however,  not  having  voluntary  motion,  cannot 
make  this  compensation  for  the  disturbing  force  which  is  call- 
ed into  existence  by  the  gradual  change  of  direction  of  the 
motion;  consequently  it  will,  under  certain  circumstances,  be 
overturned,  falling,  of  course,  outwards,  or  from  the  corner.  If 
A  B  be  the  carriage,  and  C,  jig  31.,  the  place  at  which  the 
weight  is  principally  collected,  this  point  C  will  be  under  the 
influence  of  two  forces;  the  weight,  which  may  be  represent- 
ed by  the  perpendicular  C  D,  and  the  centrifugal  force,  which 
will  be  represented  by  a  line  C  F,  which  shall  have  the  same 
proportion  to  C  D  as  the  centrifugal  force  has  to  the  weight. 
Now  the  combined  effect  of  these  two  forces  will  be  the  same 
as  the  effect  of  a  single  foi  ce,  represented  by  C  G.  Thus 
the  pressure  of  the  carriage  on  the  road  is  brought  nearer  te 


88  THE    ELEMENTS    OF    MECHANICS.  CHAP.  VIII. 

the  outer  wheel  B.  If  the  centrifugal  force  bear  the  same 
proportion  to  the  weight  as  C  F  (or  D  B),Jig.  32.,  bears  to 
C  D,  the  whole  pressure  is  thrown  upon  the  wheel  B. 

If  the  centrifugal  force  bear  to  the  weight  a  greater  pro- 
portion than  D  B  has  to  C  D,  then  the  line  C  F,  which  repre- 
sents it,  Jig.  33.,  will  be  greater  than  D  B.  The  diagonal 
C  G,  which  represents  the  combined  effects  of  the  weight  and 
centrifugal  force,  will  in  this  case  pass  outside  the  wheel  B, 
and  therefore  this  resultant  will  be  unresisted.  To  perceive 
how  far  it  will  tend  to  overturn  the  carriage,  let  the  force  C  G 
be  resolved  into  two,  one  in  the  direction  of  C  B,  and  the 
other  C  K,  perpendicular  to  C  B.  The  former  C  B  will  be 
resisted  by  the  road,  but  the  latter  C  K  will  tend  to  lift  the 
carriage  over  the  external  wheel.  If  the  velocity  and  the 
curvature  of  the  course  be  continued  for  a  sufficient  time  to 
enable  this  force  C  K  to  elevate  the  weight,  so  that  the  line  of 
direction  shall  fall  on  B,  the  carriage  will  be  overthrown. 

It  is  evident  from  what  has  been  now  stated,  that  the 
chances  of  overthrow  under  these  circumstances  depend  on 
the  proportion  of  B  D  to  C  D,  or,  what  is  to  the  same  purpose, 
of  the  distance  between  the  wheels  to  the  height  of  the  prin- 
cipal seat  of  the  load.  It  will  be  shown  in  the  next  chapter, 
that  there  is  a  certain  point,  called  the  centre  of  gravity,  at 
which  the  entire  weight  of  the  vehicle  and  its  load  may  be 
conceived  to  be  concentrated.  This  is  the  point  which  in  the 
present  investigation  we  have  marked  C.  The  security  of 
the  carriage,  therefore,  depends  on  the  greatness  of  the  dis- 
tance between  the  wheels,  and  the  smallncss  of  the  elevation 
of  the  centre  of  gravity  above  the  road ;  for  either  or  both  of 
these  circumstances  will  increase  the  proportion  of  B  D  to  C  D. 

(143.)  In  the  equestrian  feat  exhibited  in  the  ring  at  the 
amphitheatre,  when  the  horse  moves  round  with  the  performer 
standing  on  the  saddle,  both  the  horse  and  rider  incline  con- 
tinually towards  the  centre  of  the  ring,  and  the  inclination 
increases  with  the  velocity  of  the  motion  :  by  this  inclination 
their  weights  counteract  the  effect  of  the  centrifugal  force, 
exactly  as  in  the  case  already  mentioned  (142). 

(144.)  If  a  body  be  allowed  to  fall  by  its  weight  down  a 
convex  surface,  such  as  A  B,.fiff.  34.,  it  would  continue  upon 
the  surface  until  it  arrive  at  B,  but  for  the  effect  of  the  cen- 
trifugal force  :  this,  giving  it  a  motion  from  the  centre  of  the 
curve,  will  cause  it  to  quit  the  curve  at  a  certain  point  C, 
M  liich  can  be  easily  found  by  mathematical  computation. 


CHAP.    VIII.  FAMILIAR     EXAMPLES. 

(145.)  The  most  remarkable  and  important  manifestation 
of  centrifugal  force  is  observed  in  the  effects  produced  by 
the  rotation  of  the  earth  upon  its  axis.  Let  the  circle  in 
Jig.  35.  represent  a  section  of  the  earth,  A  B  being  the  axis 
on  which  it  revolves.  This  rotation  causes  the  matter  which 
composes  the  mass  of  the  earth,  to  revolve  in  circles  round 
the  different  points  of  the  axis  as  centres  at  the  various  dis- 
tances at  which  the  component  parts  of  this  mass  are  placed. 
As  they  all  revolve  with  the  same  angular  velocity,  they  will 
be  affected  by  centrifugal  forces,  which  will  be  greater  or  less 
in  proportion  as  their  distances  from  the  centre  are  greater 
or  less.  Consequently  the  parts  of  the  earth  which  are 
situated  about  the  equator,  D,  will  be  more  strongly  af- 
fected by  centrifugal  force  than  those  about  the  poles,  A  B. 
The  effect  of  this  difference  has  been  that  the  component 
matter  about  the  equator  has  actually  been  driven  farther 
from  the  centre  than  that  about  the  poles,  so  that  the  figure 
of  the  earth  has  swelled  out  at  the  sides,  and  appears  propor- 
tionally depressed  at  the  top  and  bottom,  resembling  the 
shape  of  an  orange.  An  exaggerated  representation  of  this 
figure  is  given  in  Jig.  36. ;  the  real  difference  between  the 
distances  of  the  poles  and  equator  from  the  centre  being  too 
small  to  be  perceptible  in  a  diagram.  The  exact  proportion 
of  C  A  to  C  D  has  never  yet  been  certainly  ascertained. 
Some  observations  make  C  D  exceed  C  A  by  ^T,  and 
others  by  only  ^^-.  The  latter,  however,  seems  the  more 
probable.  It  may  be  considered  to  be  included  between 
these  limits. 

The  same  cause  operates  more  powerfully  in  other  plan- 
ets which  revolve  more  rapidly  on  their  axes.  Jupiter  and 
Saturn  have  forms  which  are  considerably  more  elliptical. 

(14G.)  The  centrifugal  force  of  the  earth's  rotation  also 
affects  detached  bodies  on  its  surface.  If  such  bodies  were 
not  held  upon  the  surface  by  the  earth's  attraction,  they 
would  be  immediately  flung  off  by  the  whirling  motion  in 
which  they  participate.  The  centrifugal  force,  however, 
really  diminishes  the  effects  of  the  earth's  attraction  on  those 
bodies,  or,  what  is  the  same,  diminishes  their  weights.  If 
the  earth  were  not  revolving  on  its  axis,  the  weight  of  bodies 
in  all  places  equally  distant  from  the  centre  would  be  the 
same ;  but  this  is  not  so  when  the  bodies,  as  they  do,  move 
round  with  the  earth.  They  acquire  from  the  centrifugal 
force  a  tendency  to  fly  from  the  axis,  which  increases  with 
8* 


90  THE  ELEMENTS  OF  MECHANICS.     CHAP.  IX. 

their  distance  from  that  axis,  and  is,  therefore,  greater  the 
nearer  they  are  to  the  equator,  and  less  as  they  approach 
the  pole.  But  there  is  another  reason  why  the  centrifugal 
force  is  more  efficient  in  the  opposition  which  it  gives  to 
gravity  near  the  equator  than  near  the  poles.  This  force 
does  not  act  from  the  centre  of  the  earth,  hut  is  directed 
from  the  earth's  axis.  It  is,  therefore,  not  directly  opposed 
to  gravity,  except  on  the  equator  itself.  On  leaving  the 
equator,  and  proceeding  towards  the  poles,  it  is  less  and  less 
opposed  to  gravity,  as  will  be  plain  on  inspecting  Jig.  35., 
where  the  lines  P  C  all  represent  the  direction  of  gravity, 
and  the  lines  P  F  represent  the  direction  of  the  centrifugal 
force. 

Since,  then,  as  we  proceed  from  the  equator  towards  the 
poles,  not  only  the  amount  of  the  centrifugal  force  is  con- 
tinually diminished,  but  also  it  acts  less  and  less  in  opposition 
to  gravity,  it  follows  that  the  weights  of  bodies  are  most 
diminished  by  it  at  the  equator,  and  less  so  towards  the  poles. 

Since  bodies  are  commonly  weighed  by  balancing  them 
against  other  bodies  of  known  weight,  it  may  be  asked,  how 
the  phenomena  we  have  been  just  describing  can  be  ascer- 
tained as  a  matter  of  fact;  for  whatever  be  the  body  against 
which  it  may  be  balanced,  that  body  must  suffer  just  as  much 
diminution  of  weight  as  every  other,  and,  consequently, -all 
being  diminished  in  the  same  proportion,  the  balance  will 
be  preserved  tliough  the  weights  be  changed. 

To  render  this  effect  observable,  it  will  be  necessary  to 
compare  the  effects  of  gravity  with  some  phenomenon  which 
is  not  affected  by  the  centrifugal  force  of  the  earth's  rotation, 
and  which  will  be  the  same  at  every  part  of  the  earth.  The 
means  of  accomplishing  this  will  be  explained  in  a  subse- 
quent chapter. 


CHAPTER  IX. 

THE    CENTRE    OF    GRAVITY. 


(147.)  BY  the  earth's  attraction,  all  the  particles  which 
compose  the  mass  of  a  body  are  solicited  by  equal  forces  in 
parallel  directions  downwards.  If  these  component  particles 
were  placed  in  mere  juxtaposition,  without  any  mechanical 


CHAP.  IX.  CENTRE  OF  GRAVITY.  91 

connection,  the  force  impressed  on  any  one  of  them  could 
in  nowise  affect  the  others,  and  the  mass  would,  in  such  a 
case,  be  contemplated  as  an  aggregation  of  small  particles  of 
matter,  each  urged  by  an  independent  force.  But  the  bodies 
which  are  the  subjects  of  investigation  in  mechanical  sci- 
ence are  not  found  in  this  state.  Solid  bodies  are  coherent 
masses,  the  particles  of  which  are  firmly  bound  together,  so 
that  any  force  which  affects  one,  being  modified  according  to 
circumstances,  will  be  transmitted  through  the  whole  body. 
Liquids  accommodate  themselves  to  the  shape  of  the  surfaces 
on  which  they  rest,  and  forces  affecting  any  one  part  are 
transmitted  to  others,  in  a  manner  depending  on  the  peculiar 
properties  of  this  class  of  bodies. 

As  all  bodies,  which  are  subjects  of  mechanical  inquiry, 
on  the  surface  of  the  earth,  must  be  continually  influenced 
by  terrestrial  gravity,  it  is  desirable  to  obtain  some  easy  and 
summary  method  of  estimating  the  effect  of  this  force.  To 
consider  it,  as  is  unavoidable  in  the  first  instance,  the  com- 
bined action  of  an  infinite  number  of  equal  and  parallel 
forces  soliciting  the  elementary  molecules  downwards,  would 
be  attended  with  manifest  inconvenience.  An  infinite  num- 
ber of  forces,  and  an  infinite  subdivision  of  the  mass,  would 
form  parts  of  every  mechanical  problem. 

To  overcome  this  difficulty;  and  to  obtain  all  the  ease  and 
simplicity  which  can  be  desired  in  elementary  investigations, 
it  is  only  necessary  to  determine  some  force,  whose  single 
effect  shall  be  equivalent  to  the  combined  effects  of  the  grav- 
itation of  all  the  molecules  of  the  body.  If  this  can  be 
accomplished,  that  single  force  might  be  introduced  into  all 
problems  to  represent  the  whole  effect  of  the  earth's  attrac- 
tion, and  no  regard  need  be  had  to  any  particles  of  the 
body,  except  that  on  which  this  force  acts. 

(148.)  To  discover  such  a  force,  if  it  exist,  we  shall  first 
inouire  what  properties  must  necessarily  characterize  it.  Let 
A  B,Jig.  37.,  be  a  solid  body  placed  near  the  surface  of  the 
earth.  Its  particles  are  all  solicited  downwards,  in  the  direc- 
tions represented  by  the  Tirrows.  Now,  if  there  be  any  single 
force  equivalent  to  these  combined  effects,  two  properties 
maybe  at  once  assigned  to  it:  1.  It  must  be  presented 
downwards,  in  the  common  direction  of  those  forces  to  which 
it  is  mechanically  equivalent ;  and,  2."  it  must  be  equal  in 
intensity  to  their  sum,  or,  what  is  the  same,  to  the  force  with 
which  the  whole  mass  would  descend.  We  shall  then  sup- 


92  THE    ELEMENTS    OF    MECHANICS.  CHAP.    IX. 

pose  it  to  have  this  intensity,  and  to  have  the  direction  of  the 
arrow  D  E.  Now,  if  the  single  force,  in  the  direction  D  E, 
be  equivalent  to  all  the  separate  attractions  which  affect  the 
particles,  we  may  suppose  all  these  attractions  removed,  and 
the  body  A  B  influenced  only  by  a  single  attraction,  acting 
in  the  direction  D  E.  This  being  admitted,  it  follows  that  if 
the  body  be  placed  upon  a  prop,  immediately  under  the  direc- 
tion of  the  line  D  E,  or  be  suspended  from  a  fixed  point 
immediately  above  its  direction,  it  will  remain  motionless. 
For  the  whole  attracting  force  in  the  direction  D  E  will,  in 
the  one  case,  press  the  body  on  the  prop,  and,  in  the  other 
case,  will  give  tension  to  the  cord,  rod,  or  whatever  other 
means  of  suspension  be  used. 

(149.)  But  suppose  the  body  were  suspended  from  some 
point  P,  not  in  the  direction  of  the  line  D  E.  Let  P  C  be 
the  direction  of  the  thread  by  which  the  body  is  suspended. 
Its  whole  weight,  according  to  the  supposition  which  we  have 
adopted,  must  then  act  in  the  direction  C  E.  Taking  C  F 
to  represent  the  weight,  it  may  be  considered  as  mechani- 
cally equivalent  to  two  forces  (74),  C  I  and  C  H.  Of  these, 
C  II,  acting  directly  from  the  point  P,  merely  produces  pres- 
sure upon  it,  and  gives  tension  to  the  cord  P  C;  but  C  I, 
acting  at  right  angles  to  C  P,  produces  motion  round  P  as 
a  centre,  and  in  the  direction  C  1,  towards  a  vertical  line 
P  G,  drawn  through  the  point  P.  If  the  body  A  B  had 
been  on  the  other  side  of  .the  line  P  G,  it  \.ould  have  moved, 
in  like  manner,  towards  it,  and,  therefore,  in  the  direction 
contrary  to  its  present  motion. 

Hence  we  must  infer,  that,  when  the  body  is  suspended 
from  a  fixed  point,  it  cannot  remain  at  rest,  if  that  fixed 
point  be  not  placed  in  the  direction  of  the  line  D  E;  and, 
on  the  other  hand,  that  if  the  fixed  point  be  in  the  direction 
of  that  line,  ite  cannot  move.  A  practical  test  is  thus  sug- 
gested, by  which  the  line  D  E  may  be  at  once  discovered. 
Let  a  thread  be  attached  to  any  point  of  the  body,  and  let  it 
be  suspended  by  this  thread  from  a  hook  or  other  fixed  point. 
The  direction  of  the  thread,  when  the  body  becomes  quies- 
cent, will  be  that  of  a  single  force  equivalent  to  the  gravita- 
tion of  all  the  component  parts  of  the  mass. 

(150.)  An  inquiry  is  here  suggested  :  Does  the  direction 
of  the  equivalent  force,  thus  determined,  depend  on  the 
position  of  the  body  with  respect  to  the  surface  of  the  earth, 
and  how  is  the  direction  of  the  equivalent  force  affected  by  a 


i 


CHAP.  I*.  CENTRE  OF  GRAVITY.  93 

change  in  that  position?  This  question  may  be  at  once 
solved  if  the  body  be  suspended  by  different  points,  and  the 
directions  which  the  suspending  thread  takes  in  each  case 
relatively  to  the  figure  and  dimensions  of  the  body  ex- 
amined. 

The  body  being  suspended  in  this  manner  from  any  point, 
let  a  small  hole  be  bored  through  it,  in  the  exact  direction  of 
the  thread,  so  that,  if  the  thread  were  continued  below  th 
point  where  it  is  attached  to  the  body,  it  would  pass  throug 
this  hole.  The  body  being  successively  suspended  by  severa. 
different  points  on  its  surface,  let  as  many  small  holes  be 
bored  through  it  in  the  same  manner.  If  the  body  be  then 
cut  through,  so  as  to  discover  the  directions  which  the  several 
holes  have  taken,  they  will  be  all  found  to  cross  each  other 
at  one  point  within  the  body ;  or  the  same  fact  may  be  dis- 
covered thus :  a  thin  wire,  which  nearly  fills  the  holes,  being 
passed  through  any  one  of  them,  it  will  be  found  to  intercept 
the  passage  of  a  similar  wir,;  through  any  other. 

This  singular  fact  teaches  us,  what,  indeed,  can  be  proved 
by  mathematical  reasoning  without  experiment,  that  there  is 
one  point  in  every  body  through  which  the  single  force,  which 
is  equivalent  to  the  gravitation  of  all  its  particles,  must  pass 
in  whatever  position  the  body  be  placed.  This  point  is  called 
the  centre  of  gravity. 

(151.)  In  whatever  situation  a  body  may  be  placed,  the 
centre  of  gravity  will  have  a  tendency  to  descend  in  the  di- 
rection of  a  line  perpendicular  to  the  horizon,  and  which  is 
called  the  line  of  direction  of  the  weight.  If  the  body  be 
altogether  free  and  unrestricted  by  any  resistance  or  impedi- 
ment, the  centre  of  gravity  will  actually  descend  in  this  di- 
rection, and  all  the  other  points  of  the  body  will  move  with 
the  same  velocity  in  parallel  directions,  so  that,  during  its  fall, 
the  position  of  the  parts  of  the  body,  with  respect  to  the 
ground,  will  be  unaltered.  But  if  the  body,  as  is  most  usual, 
be  subject  to  some  resistance  or  restraint,  it  will  either  remain 
unmoved,  its  weight  being  expended  in  exciting  pressure  on 
the  restraining  points  or  surfaces,  or  it  will  move  in  a  direc- 
tion and  with  a  velocity  depending  on  the  circumstances 
which  restrain  it. 

In  order  to  determine  those  effects,  to  predict  the  pressure 
produced  by  the  weight,  if  the  body  be  quiescent,  or  the 
mixed  effects  of  motion  and  pressure,  if  it  be  not  so,  it  is 
necessary  in  all  cases  to  be  able  to  assign  the  place  of  the 


94  THE    ELEMENTS    OF    MECHANICS.  CHAP.    IX. 

centre  of  gravity.  When  the  magnitude  and  figure  of  the 
body,  and  the  density  of  the  matter  which  occupies  its  di- 
mensions, are  known,  the  place  of  the  centre  of  gravity  can 
be  determined  with  the  greatest  precision  by  mathematical 
calculation.  The  process  by  which  this  is  accomplished, 
however,  is  not  of  a  sufficiently  elementary  nature  to  be 
properly  introduced  into  this  treatise.  To  render  it  intelligi- 
ble would  require  the  aid  of  some  of  the  most  advanced 
analytical  principles  ;  and  even  to  express  the  position  of  the 
point  in  question,  except  in  very  particular  instances,  would 
be  impossible,  without  the  aid  of  peculiar  symbols. 

(152.)  There  are  certain  particular  forms  of  body  in  which, 
when  they  are  uniformly  dense,  the  place  of  the  centre  of 
gravity  can  be  easily  assigned,  and  proved  by  reasoning 
which  is  generally  intelligible ;  but  in  all  cases  whatever, 
this  point  may  be  easily  determined  by  experiment. 

(153.)  If  a  body  uniformly  dense  have  such  a  shape  that  a 
point  may  be  found,  on  either  side  of  which,  in  all  directions 
around  it,  the  materials  of  the  body  are  similarly  distributed, 
that  point  will  obviously  be  the  centre  of  gravity.  For  if  it 
be  supported,  the  gravitation  of  the  particles  on  one  side 
drawing  them  downwards,  is  resisted  by  an  effect  of  exactly 
the  same  kind  and  of  equal  amount  on  the  opposite  side,  and 
so  the  body  remains  balanced  on  the  point. 

The  most  remarkable  body  of  this  kind  is  a  globe,  the 
centre  of  which  is  evidently  its  centre  of  gravity. 

A  figure,  such  as  Jig.  38.,  called  an  oblate  spheroid,  has 
its  centre  of  gravity  at  its  centre,  C.  Such  is  the  figure  of 
the  earth.  The  same  may  be  observed  of  the  elliptical  solid, 
jig.  39.,  which  is  called  a  prolate  spheroid.  . 

A  cube,  and  some  other  regular  solids,  bounded  by  plane 
surfaces,  have  a  point  within  them,  such  as  above  described, 
and  which  is,  therefore,  their  centre  of  gravity.  Such  are 
Jig.  40. 

A  straight  wand,  of  uniform  thickness,  has  its  centre  of 
gravity  at  the  centre  of  its  length ;  and  a  cylindrical  body 
has  its  centre  of  gravity  in  its  centre,  at  the  middle  of  its 
length  or  axis.  Such  is  the  point  C,  Jig.  41. 

A  flat  plate  of  any  uniform  substance,  and  which  has,  in 
every  part,  an  equal  thickness,  has  its  centre  of  gravity  at 
the  middle  of  its  thickness,  and  under  a  point  of  its  surface, 
which  is  to  be  determined  by  its  shape.  If  it  be  circular  or 
elliptical,  this  point  is  its  centre.  If  it  have  any  regular 


CHAP.  IX.  CENTRE  OF  GRAVITY,  95 

form,  bounded  by  straight  edges,  it  is  that  point  which  is 
equally  distant  from  its  several  angles,  as  C  in  Jig.  42. 

(154.)  There  are  some  cases  in  which,  although  the  place 
of  the  centre  of  gravity  is  not  so  obvious  as  in  the  examples 
j'ust  given,  still  it  may  be  discovered,  without  any  mathemat- 
ical process,  which  is  not  easily  understood.  Suppose  ABC, 
Jig.  43.,  to  be  a  flat  triangular  plate  of  uniform  thickness 
and  density.  Let  it  be  imagined  to  be  divided  into  narrow 
bars,  by  lines  parallel  to  the  side  A  C,  as  represented  in  the 
figure.  Draw  B  D  from  the  angle  B  to  the  middle  point  D 
ef  the  side  A  C.  It  is  not  difficult  to  perceive,  that  B  D 
will  divide  equally  all  the  bars  into  which  the  triangle  is  con- 
ceived to  be  divided.  Now,  if  the  flat  triangular  plate  ABC 
be  placed  in  a  horizontal  position  on  a  straight  edge  coincid- 
ing with  the  line  B  D,  it  will  be  balanced ;  for  the  bars 
parallel  to  A  C  will  be  severally  balanced  by  the  edge  imme- 
diately under  their  middle  point,  since  that  middle  point  is 
the  centre  of  gravity  of  each  bar.  Since,  then,  the  triangle 
is  balanced  on  the  edge,  the  centre  of  gravity  must  be  some- 
where immediately  over  it,  and  must,  therefore,  be  within 
the  plate,  at  some  point  under  the  line  B  D. 

The  same  reasoning  will  prove  that  the  centre  of  gravity 
of  the  plate  is  under  the  line  A  E,  drawn  from  the  angle  A 
to  the  middle  point  E  of  the  side  B  C.  To  perceive  this,  it 
is  only  necessary  to  consider  the  triangle  divided  into  bars 
parallel  to  B  C,  and  thence  to  show  that  it  will  be  balanced 
on  an  edge  placed  under  A  E.  Since,  then,  the  centre  of 
gravity  of  the  plate  is  under  the  line  B  D,  and  also  under 
A  E,  it  must  be  under  the  point  G,  at  which  these  lines  cross 
each  other;  and  it  is  accordingly  at  a  depth  beneath  G, 
equal  to  half  the  thickness  of  the  plate. 

This  may  be  experimentally  verified  by  taking  a  piece  of 
tin  or  card,  and  cutting  it  into  a  triangular  form.  The  point 
G  being  found  by  drawing  B  D  and  A  E,  which  divide  two 
sides  equally,  it  will  be  balanced  if  placed  upon  the  point  of 
a  pin  at  G. 

The  centre  of  gravity  of  a  triangle  being  thus  determined, 
we  shall  be  able  to  find  the  position  of  the  centre  of  gravity 
of  any  plate  of  uniform  thickness  and  density  which  is 
bounded  by  straight  edges,  as  will  be  shown  hereafter  (173). 

(155.)  The  centre  of  gravity  is  not  always  included  within 
the  volume  of  the  body,  that  is,  it  is  not  enclosed  by  its  sur- 
faces. Numerous  examples  of  this  can  be  produced.  If  a 


96  THE    ELEMENTS    OF    MECHANICS.  CHAP.    IX 

piece  of  wire  be  bent  into  any  form,  the  centre  of  gravity 
will  rarely  be  in  the  wire.  Suppose  it  be  brought  to  the 
form  of  a  ring.  In  that  case,  the  centre  of  gravity  of  the 
wire  will  be  the  centre  of  the  circle,  a  point  not  forming  any 
part  of  the  wire  itself:  nevertheless  this  point  may  be  proved 
to  have  the  characteristic  property  of  the  centre  of  gravity  ; 
for  if  the  ring  be  suspended  by  any  point,  the  centre  of  the 
ring  must  always  settle  itself  under  the  point  of  suspension. 
If  this  centre  could  be  supposed  to  be  connected  with  the 
ring  by  very  fine  threads,  whose  weight  would  be  insignifi- 
cant, and  which  might  be  united  by  a  knot  or  otherwise  at 
the  centre,  the  ring  would  be  balanced  upon  a  point  placed 
under  the  knot. 

In  like  manner,  if  the  wire  be  formed  into  an  ellipse,  or 
any  other  curve  similarly  arranged  round  a  centre  point,  that 
point  will  be  its  centre  of  gravity. 

(156.)  To  find  the  centre  of  gravity  experimentally,  the 
method  described  in  (149,  150)  may  be  used.  In  this  case 
two  points  of  suspension  will  be  sufficient  to  determine  it ; 
for  the  directions  of  the  suspending  cord,  being  continued 
through  the  body,  will  cross  each  other  at  the  centre  of  grav- 
ity. These  directions  may  also  be  found  by  placing  the 
body  on  a  sharp  point,  and  adjusting  it  so  as  to  be  balanced 
upon  it.  In  this  case,  a  line  drawn  through  the  body  directly 
upwards  from  the  point  will  pass  through  the  centre  of  grav- 
ity, and,  therefore,  two  such  lines  must  cross  at  that  point. 

(157.)  If  the  body  have  two  flat  parallel  surfaces,  like 
sheet  metal,  stiff  paper,  card,  board,  &c.,  the  centre  of  grav- 
ity may  be  found  by  balancing  the  body  in  two  positions  on 
an  horizontal  straight  edge.  The  point  where  the  lines 
marked  by  the  edge  cross  each  other  will  be  immediately 
under  the  centre  of  gravity.  This  may  be  verified  by  show- 
ing that  the  body  will  be  balanced  on  a  point  thus  placed,  or 
that,  if  it  be  suspended,  the  point  thus  determined  will  always 
come  under  the  point  of  suspension. 

The  position  of  the  centre  of  gravity  of  such  bodies  may 
also  be  found  by  placing  the  body  on  an  horizontal  table 
having  a  straight  edge.  The  body  being  moved  beyond  the 
edge  until  it  is  in  that  position  in  which  the  slightest  distur- 
bance will  cause  it  to  fall,  the  centre  of  gravity  will  then 
be  immediately  over  the  edge.  This  being  done  in  two 
positions,  the  centre  of  gravity  will  be  determined  as  before. 

(158.)  It  has  been  already  stated,  that  when  the  body  is 


U1AP.    IX.  CENTRE    OP    GRAVITY.  97 

perfectly  free,  the  centre  of  gravity  must  necessarily  move 
downwards,  in  a  direction  perpendicular  to  an  horizontal 
plane.  When  the  body  is  not  free,  the  circumstances  which 
restrain  it  generally  permit  the  centre  of  gravity  to  move  in 
certain  directions,  but  obstruct  its  motion  in  others.  Thus, 
if  a  body  be  suspended  from  a  fixed  point  by  a  flexible  cord, 
the  centre  of  gravity  is  free  to  move  in  every  direction  except 
those  which  would  carry  it  farther  from  the  point  of  suspen- 
sion than  the  length  of  the  cord.  Hence  if  we  conceive  a 
globe  or  sphere  to  surround  the  point  of  suspension  on  every 
side  to  a  distance  equal  to  that  of  the  centre  of  gravity  from 
the  point  of  suspension,  when  the  cord  is  fully  stretched,  the 
centre  of  gravity  will  be  at  liberty  to  move  in  every  direction 
within  this  sphere. 

There  are  an  infinite  variety  of  circumstances  under 
which  the  motion  of  a  body  may  be  restrained,  and  in 
which  a  most  important  and  useful  class  of  mechanical  prob- 
lems originate.  Before  we  notice  others,  we  shall,  however, 
examine  that  which  has  just  been  described  more  particularly. 

Let  P,  jig.  44.,  be  the  point  of  suspension,  and  C  the 
centre  of  gravity,  and  suppose  the  body  so  placed  that  C 
shall  be  within  the  sphere  already  described.  The  cord  will 
therefore  be  slackened,  and  in  this  state  the  body  will  be 
free.  The  centre  of  gravity  will  therefore  descend  in  the 
perpendicular  direction  until  the  cord  becomes  fully  extend- 
ed ;  the  tension  will  then  prevent  its  further  motion  in  the 
perpendicular  direction.  The  downward  force  must  now  be 
considered  as  the  diagonal  of  a  parallelogram,  and  equivalent 
to  two  forces  C  D  and  C  E,  in  the  directions  of  the  sides,  as 
already  explained  in  (149).  The  force  C  D  will  bring  the 
centre  of  gravity  into  the  direction  P  F,  perpendicularly 
under  the  point  of  suspension.  Since  the  force  of  gravity 
acts  continually  on  C  in  its  approach  to  P  F,  it  will  move 
towards  that  line  with  accelerated  speed,  and  when  it  has 
arrived  there,  it  will  have  acquired  a  force  to  which  no  ob- 
struction is  immediately  opposed,  and,  consequently,  by  its 
inertia,  it  retains  this  force,  and  moves  beyond  P  F  on  the 
other  side.  But  when  the  point  C  gets  into  the  line  P  F,  it 
is  in  the  lowest  possible  position  ;  for  it  is  at  the  lowest  point 
of  the  sphere  which  limits  its  motion.  When  it  passes  to 
the  other  side  of  P  F,  it  must  therefore  begin  to  ascend,  and 
the  force  of  gravity,  which,  in  the  former  case,  accelerated 
its  descent,  will  now,  for  the  same  reason,  and  with  equal 
9 


98  THE    ELEMENTS    OF    MECHANICS.  CHAP.    IX. 

energy,  oppose  its  ascent.  This  will  be  easily  understood. 
Let  C'  be  any  point  which  it  may  have  attained  in  ascending  ; 
C'  G',  the  force  of  gravity,  is  now  equivalent  to  C'  D' 
and  C'  E'.  The  latter,  as  before,  produces  tension  ;  but  the 
former  C'  D'  is  in  a  direction  immediately  opposed  to  the 
motion,  and  therefore  retards  it.  This  retardation  will  con- 
tinue until  all  the  motion  acquired  by  the  body  in  its  descent 
from  the  first  position  has  been  destroyed,  and  then  it  will 
begin  to  return  to  P  F,  and  so  it  will  continue  to  vibrate 
from  the  one  side  to  the  other  until  the  friction  on  the  point 
P,  and  the  resistance  of  the  air,  gradually  deprive  it  of  its 
motion,  and  bring  it  to  a  state  of  rest  in  the  direction  P  F. 

But  for  the  effects  of  friction  and  atmospheric  resistance, 
the  body  would  continue  for  ever  to  oscillate  equally  from 
side  to  side  of  the  line  P  F. 

(159.)  The  phenomenon  just  developed,  is  only  an  example 
of  an  extensive  class,  Whenever  the  circumstances  which 
restrain  the  body  are  of  such  a  nature  that  the  centre  of 
gravity  is  prevented  from  descending  below  a  certain  level, 
but  not,  on  the  other  hand,  restrained  from  rising  above  it, 
the  body  will  remain  at  rest  if  the  centre  of  gravity  be  placed 
at  the  lowest  limit  of  its  level ;  any  disturbance  will  cause  it 
to  oscillate  around  this  state,  and  it  cannot  return  to  a  state 
of  rest  until  friction  or  some  other  cause  have  deprived  it  of 
the  motion  communicated  by  the  disturbing  force. 

(160.)  Under  the  circumstances  which  we  have  just  de- 
scribed, the  body  could  not  maintain  itself  in  a  state  of  rest 
in  any  position  except  that  in  which  the  centre  of  gravity  is, 
at  the  lowest  point  of  the  space  in  which  it  is  free  to  move. 
This,  however,  is  not  always  the  case.  Suppose  it  were  sus- 
pended by  an  inflexible  rod  instead  ^of  a  flexible  string;  the 
centre  of  gravity  would  then  not  only  be  prevented  from 
receding  from  the  point  of  suspension,  but  also  from  ap- 
proaching it ;  in  fact,  it  would  be  always  kept  at  the  same 
distance  from  it.  Thus,  instead  of  being  capable  of  moving 
any  where  within  the  sphere,  it  is  now  capable  of  moving  on 
its  surface  only.  The  reasoning  used  in  the  last  case  may 
also  be  applied  here,  to  prove  that  when  the  centre  of  gravity 
is  on  either  side  of  the  perpendicular  P  F,  it  will  fall  towards 
P  F,  and  oscillate,  and  that,  if  it  be  placed  in  the  line  P  F,  it 
will  remain  in  equilibrium.  But  in  this  case  there  is  another 
position,  in  which  the  centre  of  gravity  may  be  placed  so  as 
to  produce  equilibrium.  If  it  be  placed  at  the  highest  point 


CHAP.    IX.       StABLE   AND   INSTAOLE    EgtJlLiimiUM.  99 

of  the  sphere  in  which  it  moves,  the  whole  force  acting  on  it 
will  then  be  directed  on  the  point  of  suspension,  perpendicu- 
larly downwards,  and  will  be  entirely  expended  in  producing 
pressure  on  that  point ;  consequently,  the  body  will  in  this 
case  be  in  equilibrium.  But  this  state  of  equilibrium  is  of  a 
character  very  different  from  that  in  which  the  centre  of  grav- 
ity was  at  the  lowest  part  of  the  sphere.  In  the  present  case, 
any  displacement,  however  slight,  of  the  centre  of  gravity, 
will  carry  it  to  a  lower  level,  and  the  force  of  gravity  will  then 
prevent  its  return  to  its  former  state,  and  will  impel  it  down- 
wards until  it  attain  the  lowest  point  of  the  sphere,  and  round 
that  point  it  will  oscillate. 

(161.)  The  two  states  of  equilibrium  which  have  been  just 
noticed,  are  called  stable  and  instable  equilibrium.  The 
character  of  the  former  is,  that  any  disturbance  of  the  state 
produces  oscillation  about  it;  but  any  disturbance  of  the  lat- 
ter state  produces  a  total  overthrow,  and  finally  causes  oscilla- 
tion around  the  state  of  stable  equilibrium. 

Let  A  B,Jig.  45.,  be  an  elliptical  board  resting  on  its  edge 
on  an  horizontal  "plane.  In  the  position  here  represented,  the 
extremity  P  of  the  lesser  axis  being  the  point  of  support,  the 
board  is  in  stable  equilibrium ;  for  any  motion  on  either  side 
must  cause  the  centre  of  gravity  C  to  ascend  in  the  directions 
C  O,  and  oscillation  will  ensue.  If,  however,  it  rest  upon  the 
smaller  end,  as  in  Jig.  46.,  the  position  would  still  be  a  state 
of  equilibrium,  because  the  centre  of  gravity  is  directly  above 
the  point  of  support ;  but  it  would  be  instable  equilibrium,  be- 
cause the  slightest  displacement  of  the  centre  of  gravity  would 
cause  it  to  descend. 

Thus  an  egg  or  a  lemon  may  be  balanced  on  the  end ;  but 
the  least  disturbance  will  overthrow  it.  On  the  contrary,  it 
will  easily  rest  on  the  side,  and  any  disturbance  will  produce 
oscillation. 

(162.)  When  the  circumstances  under  which  the  body  is 
placed  allow  the  centre  of  gravity  to  move  only  in  an  horizon- 
tal line,  the  body  is  in  a  state  which  may  be  called  neutral 
equilibrium.  The  slightest  force  will  move  the  centre  of  grav- 
ity, but  will  neither  produce  oscillation  nor  overthrow  the  body, 
as  in  the  last  two  cases. 

An  example  of  this  state  is  furnished  by  a  cylinder  placed 
upon  an  horizontal  plane.  As  the  cylinder  is  rolled  upon  the 
plane,  the  centre  of  gravity  Gyf,g.  47.,  moves  in  a  line  paral- 
lel to  the  plane  A  B,  and  distant  from  it  by  the  radius  of  the 


100  THE    ELEMENTS  OF  MECHANICS.  CHAP.    IX. 

cylinder.  The  body  will  thus  rest  indifferently  in  any  position, 
because  the  line  of  direction  always  falls  upon  a  point  P  at 
which  the  body  rests  upon  the  plane. 

If  the  plane  were  inclined,  as  in  Jig.  48.,  a  body  might  be 
so  shaped,  that,  while  it  would  roll,  the  centre  of  gravity  would 
move  horizontally.  In  this  case,  the  body  would  rest  indiffer- 
ently on  any  part  of  the  plane,  as  if  it  were  horizontal,  pro- 
vided the  friction  be  sufficient  to  prevent  the  body  from  slid- 
ing down  the  plane. 

If  the  centre  of  gravity  of  a  cylinder  happen  not  to  coincide 
with  its  centre,  by  reason  of  the  want  of  uniformity  in  the 
materials  of  which  it  is  composed,  it  will  not  be  in  a  state  of 
neutral  equilibrium  on  an  horizontal  plane,  as  in  Jig.  47.  In 
this  case,  let  G,  Jig.  49.,  be  the  centre  of  gravity.  In  the 
position  here  represented,  where  the  centre  of  gravity  is  im- 
mediately below  the  centre  C,  the  state  will  be  stable  equilib- 
rium, because  a  motion  on  either  side  would  cause  the  cen- 
tre of  gravity  to  ascend;  but  in  Jig.  50.,  whore  G  is  immedi- 
ately above  C,  the  state  is  instable  equilibrium,  because  a 
motion  on  either  side  would  cause  G  to  descend,  and  the  body 
would  turn  into  the  position  j#,».  49. 

(163.)  A  cylinder  of  this  kind  will,  under  certain  circum- 
stances, roll  up  an  inclined  plane.  Let  A  R,Jig.  51.,  be  the 
inclined  plane,  and  let  the  cylinder  be  so  placed  that  the  line 
of  direction  from  G  shall  be  above  the  point  P  at  which  the 
cylinder  rests  upon  the  plane.  The  whole  weight  of  the  body 
acting  in  the  direction  G  D  will  obviously  cause  the  cylinder 
to  roll  towards  A,  provided  the  friction  be  sufficient  to  prevent 
sliding ;  but  although  the  cylinder  in  this  case  ascends,  the 
centre  of  gravity  G  really  descends. 

When  G  is  so  placed  that  the  line  of  direction  G  D  shall 
fall  on  the  point  P,  the  cylinder  will  be  in  equilibrium,  be- 
cause its  weight  acts  upon  the  point  on  which  it  rests.  There 
are  two  cases  represented  in  Jig.  52.  and  Jig.  53.,  in  which  G 
takes  this  position.  Fig.  52.  represents  the  state  of  stable, 
and  Jig.  53.  of  instable  equilibrium. 

(164.)  When  a  body  is  placed  upon  a  base,  its  stability  de- 
pends upon  the  position  of  the  line  of  direction  and  the  height 
of  the  centre  of  gravity  above  the  base.  If  the  line  of  direc- 
tion fall  within  the  base,  the  body  will  stand  firm  ;  if  it  fall 
on  the  edge  of  the  base,  it  will  be  in  a  state  in  which  the  slight- 
est force  will  overthrow  it  on  that  side  at  which  the  line  of 
direction  falls ;  and  if  the  line  of  direction  fall  without  the 


CHAP.  IX.        STABLE  AND  INSTABLE  EQUILIBRIUM.  101 

base,  the  body  must  turn  over  that  edge  which  is  nearest  to 
the  line  of  direction. 

In  Jig.  54.  and  Jig.  55.,  the  line  of  direction  G  P  falls  with- 
in the  base,  and  it  is  obyious  that  the  body  will  stand  firm ; 
for  any  attempt  to  turn  it  over  eitker  edge  would  cause  the 
centre  of  gravity  to  ascend.  But  in  Jig.  56.  the  line  of  direc- 
tion falls  upon  the  edge,  and  if  the  body  be  turned  over,  the 
centre  of  gravity  immediately  commences  to  descend.  Until 
it  be  turned  over,  however,  the  centre  of  gravity  is  supported 
by  the  edge. 

In  jig.  57.  the  line  of  direction  falls  outside  the  base,  the 
centre  of  gravity  has  a  tendency  to  descend  from  G  towards 
A,  and  the  body  will  accordingly  fall  in  that  direction. 

(165.)  When  the  line  of  direction  falls  within  the  base,  bodies 
will  always  stand  firm,  but  not  with  the  same  degree  of  stabil- 
ity. In  general,  the  stability  depends  on  the  height  through 
which  the  centre  of  gravity  must  be  elevated  before  the  body 
can  be  overthrown.  The  greater  this  height  is,  the  greater 
in  the  same  proportion  will  be  the  stability. 

Let  B  A  C,  Jig.  58.,  be  a  pyramid,  the  centre  of  gravity 
being  at  G.  To  turn  this  over  the  edge  B,  the  centre  of 
gravity  must  be  carried  over  the  arch  G  E,  and  must  there- 
fore be  raised  through  the  height  H  E.  If,  however,  the 
pyramid  were  taller  relatively  to  its  base,  as  in  Jig.  59.,  the 
height  H  E  would  be  proportionally  less ;  and  if  the  base  were 
very  small  in  reference  to  the  height,  as  in  Jig.  60.,  the  height 
H  E  would  be  very  small,  and  a  slight  force  would  throw  it 
over  the  edge  B. 

It  is  obvious  that  the  same  observations  may  be  applied  to 
ill  figures  whatever,  the  conclusions  just  deduced  depending 
only  on  the  distance  of  the  line  of  direction  from  the  edge  of 
the  base,  and  the  height  of  the  centre  of  gravity  above  it. 

(166.)  Hence  we  may  perceive  the  principle  on  which  the 
stability  of  loaded  carriages  depends.  When  the  load  is  placed 
at  a  considerable  elevation  above  the  wheels,  the  centre  of 
gravity  is  elevated,  and  the  carriage  becomes  proportionally 
insecure.  In  coaches  for  the  conveyance  of  passengers,  the 
luggage  is  therefore  sometimes  placed  below  the  body  of  the 
coach  ;  light  parcels  of  large  bulk  may  be  placed  on  the  top 
with  impunity. 

When  the  centre  of  gravity  of  a  carriage  is  much  elevated, 
there  is  considerable  danger  of  overthrow,  if  a  corner  be  turn- 
ed sharply  and  with  a  rapid  pace  ;  for  the  centrifugal  force 
9* 


102  THE  ELEMENTS  OF  MECHANICS.  CHAP.  IX. 

then  acting  on  the  centre  of  gravity  will  easily  raise  it  through 
the  small  height  which  is  necessary  to  turn  the  carriage  over 
the  external  wheels  (142.). 

(167.)  The  same  wagon  will  have  greater  stability  when 
loaded  with  a  heavy  substance  which  occupies  a  small  space, 
euch  as  metal,  than  when  it  carries  the  same  weight  of  a  light- 
er substance,  such  as  hay ;  because  the  centre  of  gravity  in 
the  latter  case  will  be  much  more  elevated. 

If  a  large  table  be  placed  upon  a  single  leg  in  its  centre,  it 
will  be  impracticable  to  make  it  stand  firm ;  but  if  the  pillar 
on  which  it  rests  terminate  in  a  tripod,  it  will  have  the  same 
stability  as  if  it  had  three  legs  attached  to  the  points  directly 
over  the  places  where  the  feet  of  the  tripod  rest. 

(168.)  When  a  solid  body  is  supported  by  more  points  than 
one,  it  is  not  necessary  for  its  stability  that  the  line  of  direc- 
tion should  fall  on  one  of  those  points.  If  there  be  only  two 
points  of  support,  the  line  of  direction  must  fall  between 
them.  The  body  is  in  this  case  supported  as  effectually  as  if 
it  rested  on  an  edge  coinciding  with  a  straight  line  drawn  from 
one  point  of  support  to  the  other.  If  there  be  three  points  of 
support,  which  are  not  ranged  in  the  same  straight  line,  the 
body  will  be  supported  in  the  same  manner  as  it  would  be  by 
a  base  coinciding  with  the  triangle  formed  by  straight  lines 
joining  the  three  points  of  support.  In  the  same  manner, 
whatever  be  the  number  of  points  on  which  the  body  may 
rest,  its  virtual  base  will  be  found  by  «;ippo.sing  straight  lines 
drawn,  joining  the  several  points  successively.  When  the 
line  of  direction  falls  within  this  base,  the  body  will  always 
stand  firm,  and  otherwise  not.  The  degree  of  stability  is 
determined  in  the  same  manner  as  if  the  base  were  a  con- 
tinued surface. 

(169.)  Necessity  and  experience  teach  an  animal  to  adapt 
its  postures  and  motions  to  the  position  of  the  centre  of  grav- 
ity of  his  body.  When  a  man  stands,  the  line  of  direction  of 
his  weight  must  fall  within  the  base  formed  by  his  feet.  If 
A  B,  C  D,Jig.  61.,  be  the  feet,  this  base  is  the  space  A  B  D  C. 
It  is  evident,  that  the  more  his  toes  are  turned  outwards,  the 
more  contracted  the  base  will  be  in  the  direction  E  F,  and 
the  more  liable  he  will  be  to  fall  backwards  or  forwards.  Also 
the  closer  his  feet  are  together,  the  more  contracted  the  base 
will  be  in  the  direction  G  H,  and  the  more  liable  he  will  be  to 
fall  towards  either  side. 

When  a  man  walks,  the  legs  are  alternately  lifted  from  the 


CHAP.  IX.  FAMILIAR    EXAMPLES.  103 

ground,  and  the  centre  of  gravity  is  either  unsupported  or 
thrown  from  the  one  side  to  the  other.  The  body  is  also 
thrown  a  little  forward,  in  order  that  the  tendency  of  the  cen- 
tre of  gravity  to  fall  in  the  direction  of  the  toes  may  assist 
the  muscular  action  in  propelling  the  body.  This  forward 
inclination  of  the  body  increases  with  the  speed  of  the  motion. 

But  for  the  flexibility  of  the  knee-joint,  the  labor  of  walking 
would  be  much  greater  than  it  is ;  for  the  centre  of  gravity 
would  be  more  elevated  by  each  step.  The  line  of  motion  of 
the  centre  of  gravity  in  walking  is  represented  by  Jig.  62., 
and  deviates  but  little  from  a  regular  horizontal  line,  so  that 
the  elevation  of  the  centre  of  gravity  is  subject  to  very  slight 
variation.  But  if  there  were  no  knee-joint,  as  when  a  man 
has  wooden  legs,  the  centre  of  gravity  would  move  as  in  Jig. 
(v3.,  so  that  at  each  step  the  weight  of  the  body  would  be  lift- 
ed through  a  considerable  height,  and  therefore  the  labor  of 
walking  would  be  much  increased. 

If  a  man  stand  on  one  leg,  the  line  of  direction  of  his 
weight  must  fall  within  the  space  on  which  his  foot  treads. 
The  smallness  of  this  space,  compared  with  the  height  of  the 
centre  of  gravity,  accounts  ibr  the  difficulty  of  this  feat. 

The  position  of  the  centre  of  gravity  of  the  body  changes 
with  the  posture  and  position  of  the  limbs.  If  the  arm  be 
extended  from  one  side,  the  centre  of  gravity  is  brought  near- 
er to  that  side  than  it  was  when  the  arm  hung  perpendicular- 
ly. When  dancers,  standing  on  one  leg,  extend  the  other  at 
right  angles  to  it,  they  must  incline  the  body  in  the  direction 
opposite  to  that  in  which  the  leg  is  extended,  in  order  to 
bring  the  centre  of  gravity  over  the  foot  which  supports 
them. 

When  a  porter  carries  a  load,  his  position  must  be  regulated 
by  the  centre  of  gravity  of  his  body  and  the  load  taken  to- 
gether. If  he  bore  the  load  on  his  back,  the  line  of  direction 
would  pass  beyond  his  heels,  and  he  would  fall  backwards. 
To  bring  the  centre  of  gravity  over  his  feet,  he  accordingly 
leans  forward,^-.  64. 

If  a  nurse  carry  a  child  in  her  arms,  she  leans  back  for  a 
like  reason. 

When  a  load  is  carried  on  the  head,  the  bearer  stands  up- 
right, that  the  centre  of  gravity  may  be  over  his  feet.  In 
ascending  a  hill,  we  appear  to  incline  forward,  and  in  de- 
scending, to  lean  backward ;  but  in  truth  we  are  standing  up- 
right with  respect  to  a  level  plane.  This  is  necessary  to 


104  THE  ELEMENTS  OF  MECHANICS.     CHAP.  IX. 

keep  the  line  of  direction  between  the  feet,  as  is  evident  from 
fff.  65. 

A  person  sitting  on  a  chair  which  has  no  back,  cannot  rise 
from  it  without  either  stooping  forward  to  bring  the  centre  of 
gravity  over  the  feet,  or  drawing  back  the  feet  to  bring  them 
under  the  centre  of  gravity. 

A  quadruped  never  raises  both  feet  on  the  same  side  simul- 
taneously, for  the  centre  of  gravity  would  then  be  unsupport- 
ed. Let  A  B  C  D,/g\  66.,  be  the  feet.  The  base  on  which 
it  stands  is  A  B  C  D,  and  the  centre  of  gravity  is  nearly  over 
the  point  O,  where  the  diagonals  cross  each  other.  The  legs 
A  and  C  being  raised  together,  the  centre  of  gravity  is  sup- 
ported by  the  legs  B  and  D,  since  it  falls  between  them ;  and 
when  B  and  D  are  raised,  it  is,  in  like  manner,  supported  by 
the  feet  A  and  C.  The  centre  of  gravity,  however,  is  often 
unsupported  for  a  moment ;  for  the  leg  B  is  raised  from  the 
ground  before  A  comes  to  it,  as  is  plain  from  observing  the 
track  of  a  horse's  feet,  the  mark  of  A  being  upon  or  before 
that  of  B.  In  the  more  rapid  paces  of  all  animals  the  centre 
of  gravity  is  at  intervals  unsupported. 

The  feats  of  rope-dancers  are  experiments  on  the  manage- 
ment of  the  centre  of  gravity.  The  evolutions  of  the  perform- 
er are  found  to  be  facilitated  by  holding  in  his  hand  a  heavy 
pole.  His  security  in  this  case  depends,  not  on  the  centre 
of  gravity  of  his  body,  but  on  that  of  his  body  and  the  pole 
taken  together.  This  point  is  near  the  centre  of  the  pole,  so 
that,  in  fact,  he  may  be  said  to  hold  in  his  hands  the  point  on 
the  position  of  which  the  facility  of  his  feats  depends.  With- 
out the  aid  of  the  pole,  the  centre  of  gravity  would  be  within 
the  trunk  of  the  body,  and  its  position  could  not  be  adapted 
to  circumstances  with  the  same  ease  and  rapidity. 

(170.)  The  centre  of  gravity  of  a  mass  of  fluid  is  that  point 
which  would  have  the  properties  which  have  been  proved  to 
belong  to  the  centre  of  gravity  of  a  solid,  if  the  fluid  were 
solidified  without  changing  in  any  respect  the  quantity  or  ar- 
rangement of  its  parts.  This  point  also  possesses  other  prop- 
erties, in  reference  to  fluids,  which  will  be  investigated  in 
HYDROSTATICS  and  PNEUMATICS. 

(171.)  The  centre  of  gravity  of  two  bodies  separated  from 
one  another,  is  that  point  which  would  possess  the  properties 
ascribed  to  the  centre  of  gravity,  if  the  two  bodies  were 
uiiiicd  bv  an  inflexible  line,  the  weight  of  which  might  be  neg- 
lected. To  find  this  point  mathematically  is  a  very  simple 


CHAP.  IX.      CENTRE  OF  GRAVITY  OF  A  SYSTEM.  105 

problem.  Let  A  and  B,  fig.  67.,  be  the  two  bodies,  and  a 
and  b  their  centres  of  gravity.  Draw  the  right  line  a  b,  and 
divide  it  at  C,  in  such  a  manner  that  a  C  shall  have  the  same 
proportion  to  6  C  as  the  mass  of  the  body  B  has  to  the  mass 
of  the  body  A. 

This  may  easily  be  verified  experimentally.  Let  A  and  B 
be  two  bodies,  whose  weight  is  considerable,  in  comparison 
with  that  of  the  rod  a  b,  which  joins  them.  Let  a  fine  silken 
string,  with  its  ends  attached  to  them,  be  hung  upon  a  pin  ; 
and  on  the  same  pin  let  a  plumb-line  be  suspended.  In  what- 
ever position  the  bodies  may  be  hung,  it  will  be  observed 
that  the  plumb-line  will  cross  the  rod  a  b  at  the  same  point, 
and  that  point  will  divide  the  line  a  b  into  parts  a  C  arvl 
b  C,  which  are  in  the  proportion  of  the  mass  of  B  to  the 
mass  of  A. 

(172.)  The  centre  of  gravity  of  three  separate  bodies  is 
defined  in  the  same  manner  as  that  of  two,  and  may  be  found 
by  first  determining  the  centre  of  gravity  of  two,  and  then 
supposing  their  masses  concentrated  at  that  point,  so  as  to 
form  one  body,  and  finding  the  centre  of  gravity  of  that  and 
the  third. 

In  the  same  manner  the  centre  of  gravity  of  any  number 
of  bodies  may  be  determined. 

(173.)  If  a  plate  of  uniform  thickness  be  bounded  by 
straight  edges,  its  centre  of  gravity  may  be  found  by  dividing 
it  into  triangles  by  diagonal  lines,  as  in  Jig.  68.,  and,  having 
determined  by  (154)  the  centres  of  gravity  of  the  several 
triangles,  the  centre  of  gravity  of  the  whole  plate  will  be 
their  common  centre  of  gravity  found  as  above. 

(174.)  Although  the  centre  of  gravity  takes  its  name  from 
the  familiar  properties  which  it  has  in  reference  to  detached 
bodies  of  inconsiderable  magnitude,  placed  on  or  near  the  sur- 
face of  the  earth,  yet  it  possesses  properties  of  a  much  more 
general  and  not  less  important  nature.  One  of  the  most 
remarkable  of  these  is,  that  the  centre  of  gravity  of  any  num- 
ber of  separate  bodies  is  never  affected  by  the  mutual  attrac- 
tion, impact,  or  other  influence  which  the  bodies  may  trans- 
mit from  one  to  another.  This  is  a  necessary  consequence 
of  the  equality  of  action  and  reaction  explained  in  Chapter  IV. 
For  if  A  and  B,  fg.  67.,  attract  each  other,  and  change 
their  places  to  A'  B',the  space  a  a'  will  have  to  b  b1  the  same 
proportion  as  B  has  to  A,  and,  therefore,  by  what  has  just 
been  proved  (171),  the  same  proportion  as  a  C  has  to  b  C 


106  THE  ELEMENTS  OF  MECHANICS.  CHAP.  IX. 

It  follows  that  the  remainders  ul  C  and  b'  C  will  be  in  the 
proportion  of  B  to  A,  and  that  C  will  continue  to  be  the 
centre  of  gravity  of  the  bodies  after  they  have  approached  by 
their  mutual  attraction. 

Suppose,  for  example,  that  A  and  B  were  12  Ibs.  and  8  Ibs. 
respectively,  and  that  a  b  were  40  feet.  The  point  C  must 
(171)  divide  a  b  into  two  parts,  in  the  proportion  of  8  to  12, 
or  of  2  to  3.  Hence  it  is  obvious  that  a  C  will  be  16  feet, 
and  b  C  24  feet.  Now,  suppose  that  A  and  B  attract  each 
other,  and  that  A  approaches  B  through  two  feet.  Then  B 
must  approach  A  through  three  feet.  Their  distances  from 
tr  will  now  be  14  feet  and  21  feet,  which,  being  in  the  pro- 
portion of  B  to  A,  the  point  C  will  still  be  their  centre  of 
gravity. 

Hence  it  follows,  that  if  a  system  of  bodies,  placed  at  rest, 
be  permitted  to  obey  their  mutual  attractions,  although  the 
bodies  will  thereby  be  severally  moved,  yet  their  common 
centre  of  gravity  must  remain  quiescent. 

(175.)  When  one  of  two  bodies  is  moving  in  a  straight 
line,  the  other  being  at  rest,  their  common  centre  of  gravity 
must  move  in  a  parallel  straight  line.  Let  A  and  B,Jig.  69., 
be  the  centres  of  gravity  of  the  bodies,  and  let  A  move  from 
A  to  a,  B  remaining  at  rest.  Draw  the  lines  A  B  and  a  B. 
In  every  position  which  the  body  B  assumes  during  its  motion, 
the  centre  of  gravity  C  divides  the  line  joining  them  into 
parts  A  C,  B  C,  which  are  in  the  proportion  of  the  mass  B 
to  the  mass  A.  Now,  suppose  any  number  of  lines  drawn 
from  B  to  the  line  A  a ;  a  parallel  C  c  to  A  a  through  C  di- 
vides all  these  .?ines  in  the  same  proportion  ;  and  therefore, 
while  the  body  A  moves  from  A  to  a,  the  common  centre 
of  gravity  moves  from  C  to  c. 

If  both  the  bodies  A  and  B  moved  uniformly  in  straight 
lines,  the  centre  of  gravity  would  have  a  motion  compounded 
(74)  of  the  two  motions  with  which  it  would  be  affected, 
if  each  moved  while  the  other  remained  at  rest.  In  the 
same  manner,  if  there  were  three  bodies,  each  moving  uniform- 
ly in  a  straight  line,  their  common  centre  of  gravity  would 
have  a  motion  compounded  of  that  motion  which  it  would  have 
if  one  remained  at  rest  while  the  other  two  moved,  and  that 
which  the  motion  of  the  first  would  give  it  if  the  last  two 
remained  at  rest ;  and  in  the  same  manner  it  may  be  proved, 
that  when  any  number  of  bodies  move  each  in  a  straight 
line,  their  common  centre  of  gravity  will  have  a  motion  com 


CHAP.  IX.     ROTATORY  AND  PROGRESSIVE  MOTION.        107 

pounded  of  the  motions  which  it  receives  from  the  bodies 
severally. 

It  may  happen  that  the  several  motions  which  the  centre 
of  gravity  receives  from  the  hodies  of  the  system  will  neutral- 
ize each  other  ;  and  this  does,  in  fact,  take  place  for  such 
motions  as  are  the  consequences  of  the  mutual  action  of  the 
bodies  upon  one  another. 

(176.)  If  a  system  of  bodies  be  not  under  the  immediate 
influence  of  any  forces,  and  their  mutual  attraction  be  con- 
ceived to  be  suspended,  they  must  severally  be  either  at  rest 
or  in  uniform  rectilinear  motion  in  virtue  of  their  inertia. 
Hence  their  common  centre  of  gravity  must  also  be  either 
at  rest  or  in  uniform  rectilinear  motion.  Now,  if  we  suppose 
their  mutual  attractions  to  take  effect,  the  state  of  their  com- 
mon centre  of  gravity  will  not  be  changed,  but  the  bodies 
will  severally  receive  motions  compounded  of  their  previous 
uniform  rectilinear  motions  and  those  which  result  from  their 
mutual  attractions.  The  combined  effects  will  cause  each 
body  to  revolve  in  an  orbit  round  the  common  centre  of  grav- 
ity, or  will  precipitate  it  towards  that  point.  But  still  that 
point  will  maintain  its  former  state  undisturbed. 

This  constitutes  one  of  the  general  laws  of  mechanical 
science,  and  is  of  great  importance  in  physical  astronomy. 
It  is  known  by  the  title  "  the  conservation  of  the  motion  of 
the  centre  of  gravity."  { 

(177.)  The  solar  system  is  an  instance  of  the  class  of  phe, 
nomena  to  which  we  have  just  referred.  All  the  motions 
of  the  bodies  which  compose  it  can  be  traced  to  certain  uni- 
form rectilinear  motions,  received  from  some  former  impulse, 
or  from  a  force  whose  action  has  been  suspended,  and  those  mo- 
tions which  necessarily  result  from  the  principle  of  gravitation. 
But  we  shall  not  here  insist  further  on  this  subject,  which 
more  properly  belongs  to  another  department  of  the  science. 

(178.)  If  a  solid  body  suffer  an  impact  in  the  direction 
of  a  line  passing  through  its  centre  of  gravity,  all  the  parti- 
cles of  the  body  will  be  driven  forward  with  the  same  velocity 
in  lines  parallel  to  the  direction  of  the  impact,  and  the  whole 
force  of  the  motion  will  be  equal  to  that  of  the  impact.  The 
common  velocity  of  the  parts  of  the  body  will  in  this  case  be 
determined  by  the  principles  explained  in  Chapter  IV.  The 
impelling  force  being  equally  distributed  among  all  the  parts, 
the  velocity  will  be  found  by  dividing  the  numerical  value  of 
that  force  by  the  number  expressing  the  mass. 


108  THE  ELEMENTS   OF  MECHANICS.  CHAP.  X 

If  any  number  of  impacts  be  given  simultaneously  to  dif- 
ferent points  of  a  body,  a  certain  complex  motion  will  gener- 
ally ensue.  The  mass  will  have  a  relative  motion  round  the 
centre  of  gravity  as  if  it  were  fixed,  while  that  point  will 
move  forward  uniformly  in  a  straight  line,  carrying  the  body 
with  it.  The  relative  motion  of  the  mass  round  the  centre 
of  gravity  may  be  found  by  considering  the  centre  of  gravity 
as  a  fixed  point,  round  which  the  mass  is  free  to  move, 
and  then  determining  the  motion  which  the  applied  forces 
would  produce.  This  motion  being  supposed  to  continue 
uninterrupted,  let  all  the  forces  be  imagined  to  be  ap- 
plied in  their  proper  directions  and  quantities  to  the  centre 
of  gravity.  By  the  principles  for  the  composition  of  force 
they  will  be  mechanically  equivalent  to  a  single  force  through 
that  point.  In  the  direction  of  this  single  force  the  centre 
of  gravity  will  move,  and  have  the  same  velocity  as  if  the 
whole  mass  were  there  concentrated  and  received  the  impel- 
ling forces. 

(179.)  These  general  properties,  which  are  entirely  inde- 
pendent of  gravity,  render  the  "  centre  of  gravity"  an  inade- 
quate title  for  this  important  point.  Some  physical  writers 
have,  consequently,  called  it  the  "  centre  of  inertia."  The 
"  centre  of  gravity,"  however,  is  the  name  by  which  it  is  still 
generally  designated. 


CHAPTER  X. 

THE  MECHANICAL  PROPERTIES  OF  AN  AXIS. 

(180.)  WHEN  a  body  has  a  motion  of  rotation,  the  line 
round  which  it  revolves  is  called  an  axis.  Every  point  of  the 
body  must  in  this  case  move  in  a  circle,  whose  centre  lies 
in  the  axis,  and  whose  radius  is  the  distance  of  the  point  from 
the  axis.  Sometimes  while  the  body  revolves,  the  axis  itself 
is  movable,  and  not  unfrequently  in  a  state  of  actual  motion. 
The  motions  of  the  earth  and  planets,  or  that  of  a  common 
spinning-top,  are  examples  of  this.  The  cases,  however, 
which  will  be  considered  in  the  .present  chapter,  are  chiefly 
those  in  which  the  axis  is  immovable,  or  at  least  where  its 
motion  has  no  relation  to  the  phenomena  under  investigation. 
Instances  of  this  are  so  frequent  and  obvious,  that  it  seems 


CHAP.  X.  PROPERTIDS  OF  AN  AXIS.  109 

scarcely  necessary  to  particularize  them.  Wheel-work  of 
every  description,  the  moving  parts  of  watches  and  clocks, 
turning  lathes,  mill-work,  doors  and  lids  on  hinges,  are  all 
obvious  examples.  In  tools  or  other  instruments  which  work 
on  joints  or  pivots,  such  as  scissors,  shears,  pincers,  although 
the  joint  or  pivot  be  not  absolutely  fixed,  it  is  to  be  considered 
so  in  reference  to  the  mechanical  effect. 

In  some  cases,  as  in  most  of  the  wheels  of  watches  and 
clocks,  fly-wheels  and  chucks  of  the  turning  lathe,  and  the 
arms  of  wind-mills,  the  body  turns  continually  in  the  same 
direction,  and  each  of  its  points  traverses  a  complete  circle 
during  every  revolution  of  the  body  round  its  axis.  In  other 
instances,  the  motion  is  alternate  or  reciprocating,  its  direction 
being  at  intervals  reversed.  Such  is  the  case  in  pendulums 
of  clocks,  balance-wheels  of  chronometers,  the  treddle  of  the 
lathe,  doors  and  lids  on  hinges,  scissors,  shears,  pincers,  &,c. 
When  the  alternation  is  constant  and  regular,  it  is  called 
oscillation  or  vibration,  as  in  pendulums  and  balance-wheels. 

(181.)  To  explain  the  properties  of  an  axis  of  rotation,  it 
will  be  necessary  to  consider  the  different  kinds  of  forces,  to 
the  action  of  which  a  body  movable  on  such  an  axis  may  be 
submitted,  to  show  how  this  action  depends  on  their  several 
quantities  and  directions,  to  distinguish  the  cases  in  which 
the  forces  neutralize  each  other,  and  mutually  equilibrate  from 
those  in  which  motion  ensues,  to  determine  the  effect  which 
the  axis  suffers,  and,  in  the  cases  where  motion  is  produced, 
to  estimate  the  effects  of  those  centrifugal  forces  (137.)  which 
are  created  by  the  mass  of  the  body  whirling  round  the  axis. 

Forces  in  general  have  been  distinguished,  by  the  duration 
of  their  action,  into  instantaneous  and  continued  forces.  The 
effect  of  an  instantaneous  force  is  produced  in  an  infinitely 
short  time.  If  the  body  which  sustains  such  an  action  be 
previously  quiescent  and  free,  it  will  move  with  a  uniform 
velocity  in  the  direction  of  the  impressed  force.  (93.)  If, 
on  the  other  hand,  the  body  be  not  free,  but  so  restrained 
that  the  impulse  cannot  put  it  in  motion,  then  the  fixed  points 
or  lines  which  resist  the  motion  sustain  a  corresponding  shock 
at  the  moment  of  the  impulse.  This  effect,  which  is  called 
percussion,  is,  like  the  force  which  causes  it,  instantaneous. 

A  continued  force  produces  a  continued  effect.  If  the 
body  be  free  and  previously  quiescent,  this  effect  is  a  con- 
tinual increase  of  velocity.  If  the  body  be  so  restrained 
that  the  appfced  force  cannot  put  it  in  motion,  the  effect  is 
10 


110  THE  ELEMENTS  OF  MECHANICS.       CHAP.  X. 

a  continued  pressure  on  the  points  or  lines  which  sustain 
it.  (94.) 

It  may  happen,  however,  that  although  the  body  be  not 
absolutely  free  to  move  in  obedience  to  the  force  applied  to 
it,  yet  still  it  may  not  be  altogether  so  restrained  as  to  resist 
the  effect  of  that  force,  and  remain  at  rest.  If  the  point  at 
which  a  force  is  applied  be  free  to  move  in  a  certain  direction 
not  coinciding  with  that  of  the  applied  force,  that  force  will 
be  resolved  into  two  elements  ;  one  of  which  is  in  the  direc- 
tion in  which  the  point  is  free  to  move,  and  the  other  at  right 
angles  to  that  direction.  The  point  will  move  in  obedience 
to  the  former  element,  and  the  latter  will  produce  percussion 
or  pressure  on  the  points  or  lines  which  restrain  the  body. 
In  fact,  in  such  cases,  the  resistance  offered  by  the  circum- 
stances which  confine  the  motion  of  the  body  modifies  the 
motion  which  it  receives,  and,  as  every  change  of  motion  must 
be  the  consequence  of  a  force  applied  (44.),  the  fixed  points 
or  lines  which  offer  the  resistance  must  suffer  a  corresponding 
effect. 

It  may  happen  that  the  forces  impressed  on  the  body, 
whether  they  be  continued  or  instantaneous,  are  such  as, 
were  it  free,  would  communicate  to  it  a  motion  which  the 
circumstances  which  restrain  it  do  not  forbid  it  to  receive. 
In  such  a  case,  the  fixed  points  or  lines  which  restrain  the 
body  sustain  no  force,  and  the  phenomena  will  be  the  same 
in  all  respects  as  if  these  points  or  lines  were  not  fixed. 

It  will  be  easy  to  apply  these  general  reflections  to  the  case 
in  which  a  solid  body  is  movable  on  a  fixed  axis.  Such  a 
body  is  susceptible  of  no  motion  except  one  of  rotation  on 
that  axis.  If  it  be  submitted  to  the  action  of  instantaneous 
forces,  one  or  other  of  the  following  effects  must  ensue. 

1.  The  axis  may  resist  the  forces,  and  prevent  any  motion. 

2.  The  axis  may  modify  the  effect  of  the  forces  sustaining 
a  corresponding  percussion,  and  the  body  receiving  a  motion 
of  rotation.     3.  The  forces  applied  may  be  such  as  would 
cause  the  body  to  spin  round  the  axis  even  were  it  not  fixed, 
in  which  case  the  body  will  receive  a  motion  of  rotation,  but 
the  axis  will  suffer  no  percussion. 

What  has  been  just  observed  of  the  effect  of  instantaneous 
forces  is  likewise  applicable  to  continued  ones.  1.  The  axis 
may  entirely  resist  the  effect  of  such  forces,  in  which  case 
it  will  suffer  a  pressure  which  may  be  estimated  by  the  rules 
for  the  composition  of  force.  2.  It  may  modify  the  effect 


CHAP.  X.          PROPERTIES  OP  AN  AXIS.  Ill 

of  the  applied  forces,  in  which  case  it  must  also  sustain  a 
pressure,  and  the  body  must  receive  a  motion  of  rotation 
which  is  subject  to  constant  variation,  owing  to  the  incessant 
action  of  the  forces.  3.  The  forces  may  be  such  as  would 
communicate  to  the  body  the  same  rotatory  motion  if  the 
axis  were  not  fixed.  In  this  case,  the  forces  will  produce  no 
pressure  on  the  axis. 

The  impressed  forces  are  not  the  only  causes  which  affect 
the  axis  of  a  body  during  the  phenomenon  of  rotation. 
This  species  of  motion  calls  into  action  other  forces  depend- 
ing on  the  inertia  of  the  mass,  which  produce  effects  upon 
the  axis,  and  which  play  a  prominent  part  in  the  theory  of 
rotation.  While  the  body  revolves  on  its  axis,  the  component 
particles  of  its  mass  move  in  circles,  the  centres  of  which 
are  placed  in  the  axis.  The  radius  of  the  circle  in  which 
each  particle  moves  is  the  line  drawn  from  that  particle 
perpendicular  to  the  axis.  It  has  been  already  proved  that  a 
particle  of  matter,  having  a  circular  motion,  is  attended  with 
a  centrifugal  force  proportionate  to  the  radius  of  the  circle 
in  which  it  moves  and  to  the  square  of  its  angular  velocity. 
When  a  solid  body  revolves  on  its  axis,  all  its  parts  are  whirled 
round  together,  each  performing  a  complete  revolution  in  the 
same  time.  The  angular  velocity  is  consequently  the  same 
for  all,  and  the  difference  of  the  centrifugal  forces  of  differ- 
ent particles  must  entirely  depend  upon  their  distances  from 
the  axis.  The  tendency  of  each  particle  to  fly  from  the 
axis,  arising  from  the  centrifugal  force,  is  resisted  by  the 
cohesion  of  the  parts  of  the  mass,  and,  in  general,  this  ten- 
dency is  expended  in  exciting  a  pressure  or  strain  upon  the 
axis.  It  ought  to  be  recollected,  however,  that  this  pressure 
or  strain  is  altogether  different  from  that  already  mentioned, 
and  produced  by  the  forces  which  give  motion  to  the  body. 
The  latter  depends  entirely  upon  the  quantity  arid  directions 
of  the  applied  forces  in  relation  to  the  axis ;  the  former  de- 
pends on  the  figure  and  density  of  the  body,  and  the  velocity 
of  its  motion. 

These  very  complex  effects  render  a  simple  and  elementary 
exposition  of  the  mechanical  properties  of  a  fixed  axis  a 
matter  of  considerable  difficulty.  Indeed,  the  complete 
mathematical  developement  of  this  theory  long  eluded  the 
skill  of  the  most  acute  geometers;  and  it  was  only  at  a 
comparatively  late  period  that  it  yielded  to  the  searching 
analysis  of  modern  science. 


112  THE    ELEMENTS    OF    MECHANICS.  CHAP.    X. 

(182.)  To  commence  with  the  most  simple  case,  we  shall 
consider  the  body  as  submitted  to  the  action  of  a  single  force. 
The  effect  of  this  force  will  vary  according  to  the  relation  of 
its  direction  to  that  of  the  axis.  There  are  two  ways  in 
which  a  body  may  be  conceived  to  be  movable  around  an  axis. 
1.  By  having  pivots  at  two  points  which  rest  in  sockets,  so 
that,  when  the  body  is  moved,  it  must  revolve  round  the  right 
line,  joining  the  pivots  as  an  axis.  2.  A  thin  cylindrical  rod 
may  pass  through  the  body,  on  which  it  may  turn  in  the 
same  manner  as  a  wheel  upon  its  axle. 

If  the  force  be  applied  to  the  body  in  the  direction  of  the 
axis,  it  is  evident  that  no  motion  can  ensue,  and  the  effect 
produced  will  be  a  pressure  on  that  pivot  towards  which  the 
force  is  directed.  If,  in  this  case,  the  body  revolved  on  a 
cylindrical  rod,  the  tendency  of  the  force  would  be  to  make 
it  slide  along  the  rod  without  revolving  round  it. 

Let  us  next  suppose  the  force  to  be  applied,  not  in  the  di- 
rection of  the  axis  itself,  but  parallel  to  it.  Let  A  B,Jig.  70., 
be  the  axis,  and  let  C  D  be  the  direction  of  the  force  applied. 
The  pivots  being  supposed  to  be  at  A  and  B,  draw  A  G  and 
B  F  perpendicular  to  A  B.  The  force  C  D  will  be  equiva- 
lent to  three  forces,  one  acting  from  B  towards  A,  equal  in 
quantity  to  the  force  C  D.  This  force  will  evidently  produce 
a  corresponding  pressure  on  the  pivot  A.  The  other  two 
forces  will  act  in  the  directions  A  G  and  B  F,  and  will  have 
respectively  to  the  force  C  D  the  same  proportion  as  A  E  has 
to  A  B.  Such  will  be  the  mechanical  effect  of  a  force  C  D 
parallel  to  the  axis.  And  as  these  effects  are  all  directed  on 
the  pivots,  no  motion  can  ensue. 

If  the  body  revolve  on  a  cylindrical  rod,  the  forces  A  G 
and  B  F  would  produce  a  strain  upon  the  axis,  while  the 
third  force  in  the  direction  B  A  would  have  a  tendency  to 
make  the  body  slide  along  it. 

(183.)  If  the  force  applied  to  the  body  be  directed  upon 
the  axis,  and  at  right  angles  to  it,  no  motion  can  be  produced. 
In  this  case,  if  the  body  be  supported  by  pivots  at  A  and  B, 
the  force  K  L,  perpendicular  to  the  line  A  B,  will  be  distrib- 
uted between  the  pivots,  producing  a  pressure  on  each  pro- 
portional to  its  distance  from  the  other ;  the  pressure  on  A 
having  to  the  pressure  on  B  the  same  proportion  as  L  B  has 
to  L  A. 

If  the  force  K  H  be  directed  obliquely  to  the  axis,  it  will 
be  equivalent  to  two  forces  (76.),  one  K  L  perpendicular  to 


CHAP.    X.  PROPERTIES    OF    AN    AXIS.  113 

the  axis,  and  the  other  K  M  parallel  to  it.  The  effect  of 
each  of  these  may  be  investigated  as  in  the  preceding  cases. 

In  all  these  observations  the  body  has  been  supposed  to  be 
submitted  to  the  action  of  one  force  only.  If  several  forces 
act  upon  it,  the  direction  of  each  of  them  crossing  the  axis 
either  perpendicularly  or  obliquely,  or  taking  the  direction  of 
the  axis  or  any  parallel  direction,  their  effects  may  be  similar- 
ly investigated.  In  the  same  manner  we  may  determine  the 
effects  of  any  number  of  forces  whose  combined  results  are 
mechanically  equivalent  to  forces  which  either  intersect  the 
axis  or  are  parallel  to  it. 

(184.)  If  any  force  be  applied  whose  direction  lies  in  a 
plane  oblique  to  the  axis,  it  can  always  be  resolved  into  two 
elements  (76.),  one  of  which  is  parallel  to  the  axis,  and  the 
other  in  a  plane  perpendicular  to  it.  The  effect  of  the  for- 
mer has  been  already  determined,  and  therefore  we  shall  at 
present  confine  our  attention  to  the  latter. 

Suppose  the  axis  to  be  perpendicular  to  the  paper,  and  to 
pass  through  the  point  G,  fi.g.  71.,  and  let  A  B  C  be  a  section 
of  the  body.  It  will  be  convenient  to  consider  the  section 
vertical  and  the  axis  horizontal,  omitting,  however,  any  notice 
of  the  effect  of  the  weight  of  the  body. 

Let  a  weight  W  be  suspended  by  a  cord  Q,  W  from  any 
point  Q.  This  weight  will  evidently  have  a  tendency  to 
turn  the  body  round  in  the  direction  ABC.  Let  another 
cord  be  attached  to  any  other  point  P,  and,  being  carried  over 
a  wheel  R,  let  a  dish  S  be  attached  to  it,  and  let  fine  sand 
be  poured  into  this  dish  until  the  tendency  of  S  to  turn  the 
body  round  the  axis  in  the  direction  of  C  B  A  balances  the 
opposite  tendency  of  W.  Let  the  weights  of  W  and  S  be 
then  exactly  ascertained,  and  also  let  the  distances  G  I  and 
G  H  of  the  cords  from  the  axis  be  exactly  measured.  It  will 
be  found  that,  if  the  number  of  ounces  in  the  weight  S  be 
multiplied  by  the  number  of  inches  in  G  H,  and  also  the 
number  of  ounces  in  W  by  the  number  of  inches  in  G  I, 
equal  products  will  be  obtained.  This  experiment  may  be 
varied  by  varying  the  position  of  the  wheel  R,  and  thereby 
changing  the  direction  of  the  string  P  R,  in  which  cases  it 
will  be  always  found  necessary  to  vary  the  weight  of  S  in 
such  a  manner,  that  when  the  number  of  ounces  in  it  is  mul- 
tiplied by  the  number  of  inches  in  the  distance  of  the  string 
from  the  axis,  the  product  obtained  shall  be  equal  to  that  of 
the  weiorht  W  by  the  distance  G  I.  We  have  here  used 
10* 


114  THE    ELEMENTS    OF   MECHANICS.  CHAP.    X. 

ounces  and  inches  as  the  measures  of  weight  and  distance ; 
but  it  is  obvious  that  any  other  measures  would  be  equally 
applicable. 

From  what  has  been  just  stated  it  follows,  that  the  energy 
of  the  weight  of  S  to  move  the  body  on  its  axis,  docs  not  de- 
pend alone  upon  the  actual  amount  of  that  weight,  but  also 
upon  the  distance  of  the  string  from  the  axis.  If,  while  the 
position  of  the  string  remains  unaltered,  the  weight  of  S  be 
increased  or  diminished,  the  resisting  weight  W  must  be  in- 
creased or  diminished  in  the  same  proportion.  But  if  while 
the  weight  of  S  remains  unaltered,  the  distance  of  the  string 
P  R  from  the  axis  G  be  increased  or  diminished,  it  will  be 
found  necessary  to  increase  or  diminish  the  resisting  weight 
W  in  exactly  the  same  proportion.  It  therefore  appears  that 
the  increase  or  diminution  of  the  distance  of  the  direction 
of  a  force  from  the  axis  has  the  same  effect  upon  its  power 
to  give  rotation  as  a  similar  increase  or  diminution  of  the 
force  itself.  The  power  of  a  force  to  produce  rotation  is, 
therefore,  accurately  estimated,  not  by  the  force  alone,  but 
by  the  product  found  by  multiplying  the  force  by  the  distance 
of  its  direction  from  the  axis.  It  is  frequently  necessary  in 
mechanical  science  to  refer  to  this  power  of  a  force,  and, 
accordingly,  the  product  just  mentioned  has  received  a  par- 
ticular denomination.  It  is  called  the  moment  of  the  force 
round  the  axis. 

(185.)  The  distance  of  the  direction  of  a  force  from  the 
axis  is  sometimes  called  the  leverage  of  the  force.  The  mo- 
ment of  a  force  is,  therefore,  found  by  multiplying  the  force 
by  its  leverage,  and  the  energy  of  a  given  force  to  turn  a 
body  round  an  axis  is  proportional  to  the  leverage  of  that 
force. 

From  all  that  has  been  observed,  it  may  easily  be  inferred 
that,  if  several  forces  affect  a  body  movable  on  an  axis,  having 
tendencies  to  turn  it  in  different  directions,  they  will  mutu- 
ally neutralize  each  other  and  produce  equilibrium,  if  the 
sum  of  the  moments  of  those  forces  which  tend  to  turn  the 
body  in  one  direction  be  equal  to  the  sum  of  the  moments  of 
Jjiose  which  tend  to  turn  it  in  the  opposite  direction.  Thus, 
if  the  forces  A,  B,  C,  .  .  .  tend  to  turn  the  body  from  right  to 
left,  and  the  distances  of  their  directions  from  the  axis  be 
«,  b,  c,  .  .  .  and  the  forces  A',  B',  C',  .  .  .  tend  to  move  it  from 
left  to  right,  and  the  distances  of  their  directions  from  the 
axis  be  a',  b',  c',  .  .  . ;  then  these  forces  will  produce  equilib- 


CHAP.  X.         MOTION  ROUND  AN  AXIS.  115 

rium,  if  the  products  found  by  multiplying  the  ounces  in 
A,  B,  C,  .  .  .  respectively  by  the  inches  in  <z,  6,  e,  .  .  .  when 
added  together,  be  equal  to  the  products  found  by  multiplying 
the  ounces  in  A',  B',  C',  .  .  .  by  the  inches  in  a',  &',  c1,  .  . 
respectively  when  added  together.  But  if  either  of  these 
sets  of  products  when  added  together  exceed  the  other,  the 
corresponding  set  of  forces  will  prevail,  and  the  body  will 
Devolve  on  its  axis. 

(186.)  When  a  body  receives  an  impulse  in  a  direction 
perpendicular  to  the  axis,  but  not  crossing  it,  a  uniform  rotato- 
ry motion  is  produced.  The  velocity  of  this  motion  depends 
on  the  force  of  the  impulse,  the  distance  of  the  direction  of 
the  impulse  from  the  axis,  and  the  manner  in  which  the  mass 
of  the  body  is  distributed  round  the  axis.  It  is  to  be  con- 
sidered that  the  whole  force  of  the  impulse  is  shared  amongst 
the  various  parts  of  the  mass,  and  is  transmitted  to  them 
from  the  point  where  the  impulse  is  applied  by  reason  of  the 
cohesion  and  tenacity  of  the  parts,  and  the  impossibility  of 
one  part  yielding  to  a  force  without  carrying  all  the  other 
parts  with  it.  The  force  applied  acts  upon  those  particles 
nearer  to  the  axis  than  its  own  direction  under  advantageous 
circumstances ;  for,  according  to  what  has  been  already  ex- 
plained, their  power  to  resist  the  effect  of  the  applied  force 
is  small  in  the  same  proportion  with  their  distance.  On  the 
other  hand,  the  applied  force  acts  upon  particles  of  the  mass, 
at  a  greater  distance  than  its  own  direction,  under  circum- 
stances proportionably  disadvantageous;  for  their  resistance 
to  the  applied  force  is  great  in  proportion  to  their  distances 
from  the  axis. 

Let  C  D,  Jig.  72.,  be  a  section  of  the  body  by  a  plane 
passing  through  the  axis  A  B.  Suppose  the  impulse  to  be 
applied  at  P,  perpendicular  to  this  plane,  and  at  the  distance 
P  O  from  the  axis.  The  effect  of  the  impulse  being  distrib- 
uted through  the  mass  will  cause  the  body  to  revolve  on  A  B 
with  a  uniform  velocity.  There  is  a  certain  point  G,  at 
which,  if  the  whole  mass  were  concentrated,  it  would  receive 
from  the  impulse  the  same  velocity  round  the  axis.  The  dis- 
tance O  G  is  called  the  radius^of  gyration  of  the  axis  A  B, 
and  the  point  G  is  called  the  centre  of  gyration  relatively  to 
that  axis.  The  effect  of  the  impulse  upon  the  mass  concen- 
trated at  G  is  great  in  exactly  the  same  proportion  as  O  G  is 
small.  This  easily  follows  from  the  property  of  moments, 


116 


THE     ELEMENTS    OF    MECHANICS.  CHAP.    X 


which  has  been  already  explained ;  from  whence  it  may  be 
inferred,  that  the  greater  the  radius  of  gyration  is,  the  less 
willj>e  the  velocity  which  the  body  will  receive  from  a  given 
impulse. 

(187.)  Since  the  radius  of  gyration  depends  on  the  manner 
in  which  the  mass  is  arranged  round  the  axis,  it  follows  that 
for  different  axes  in  the  same  body  there  will  be  different 
radii  of  gyration.  Of  allaxes  taken  in  the  same  body  par- 
allel to  each  other,  that  which  passes  through  the  centre  of 
gravity  has  the  least  radius  of  gyration.  If  the  radius  of 
gyration  of  any  axis  passing  through  the  centre  of  gravity  l><* 
given,  that  of  any  parallel  axis  can  be  found  ;  for  the  square 
of  the  radius  of  gyration  of  any  axis  is  equal  to  the  square 
of  the  distance  of  that  axis  from  the  centre  of  gravity  added 
to  the  square  of  the  radius  of  gyration  of  the  parallel  axis 
through  the  centre  of  gravity. 

(188.)  The  product  of  the  numerical  expressions  for  the 
mass  of  the  body  and  the  square  of  the  radius  of  gyration  is 
a  quantity  much  used  in  mechanical  science,  and  has  been 
called  the  moment  of  inertia.  The  moments  of  inertia, 
therefore,  for  different  axes  in  the  same  body,  are  proportional 
to  the  squares  of  the  corresponding  radii  of  gyration,  and, 
consequently,  increase  as  the  distances  of  the  axes  from  the 
centre  of  gravity  increase  (187). 

(189.)  From  what  has  been  explained  in  (187.),  it  follows, 
that  the  moment  of  inertia  of  any  axis  may  be  computed  by 
common  arithmetic,  if  the  moment  of  inertia  of  a  parallel 
axis  through  the  centre  of  gravity  be  previously  known.  To 
determine  this  last,  however,  would  require  analytical  pro- 
cesses altogether  unsuitable  to  the  nature  and  objects  of  the 
present  treatise. 

The  velocity  of  rotation  which  a  body  receives  from  a 
given  impulse  is  great  in  exactly  the  same  proportion  as  the 
moment  of  inertia  is  small.  Thus  the  moment  of  inertia 
may  be  considered  in  rotatory  motion  analogous  to  the  mass 
of  the  body  in  rectilinear  motion. 

From  what  has  been  explained  in  (187.)  it  follows  that  a 
given  impulse  at  a  given  distance  from  the  axis  will  commu- 
nicate the  greatest  angular  velocity  when  the  axis  passes 
through  the  centre  of  gravity,  and  that  the  velocity  which  it 
will  communicate  round  other  axes  will  be  diminished  in  the 
same  proportion  as  the  squares  of  their  distances  from  the 


CHAP.    X.  PRINCIPAL    AXES.  117 

centre  of  gravity,  added  to  the  square  of  the  radius  of  gyra- 
tion for  a  parallel  axis  through  the  centre  of  gravity,  are 
augmented. 

(190.)  If  any  point  whatever  he  assumed  in  a  body,  and 
right  lines  be  conceived  to  diverge  in  all  directions  from  that 
point,  there  are  generally  two  of  these  lines,  which,  being 
taken  as  axes  of  rotation,  one  has  a  greater  and  the  other  a 
less  moment  of  inertia  than  any  of  the  others.  It  is  a  re- 
markable circumstance,  that  whatever  be  the  nature  of  the 
body,  whatever  be  its  shape,  and  whatever  be  the  position  of 
the  point  assumed,  these  two  axes  of  greatest  and  least  mo- 
ment will  always  be  at  right  angles  to  each  other. 

These  axes  and  a  third  through  the  same  point,  and  at 
right  angles  to  both  of  them,  are  called  the  principal  axes 
of  that  point  from  which  they  diverge.  To  form  a  distinct 
notion  of  their  relative  position,  let  the  axis  of  greatest  mo- 
ment be  imagined  to  lie  horizontally  from  north  to  south,  and 
the  axis  of  least  moment  from  east  to  west ;  then  the  third 
principal  axis  will  be  presented  perpendicularly  upwards  and 
downwards.  The  first  two  being  called  the  principal  axes  of 
greatest  and  least  moment,  the  third  may  be  called  the  inter- 
mediate principal  axis. 

(191.)  Although  the  moments  of  the  three  principal  axes 
be  in  general  unequal,  yet  bodies  may  be  found  having  cer- 
tain axes  for  which  these  moments  may  be  equal.  In  some 
cases,  the  moment  of  the  intermediate  axis  is  equal  to  that  of 
the  principal  axis  of  greatest  moment :  in  others,  it  is  equal  to 
that  of  the  principal  axis  of  least  moment,  and  in  others  the 
moments  of  all  the  three  principal  axes  are  equal  to  each 
other. 

If  the  moments  of  any  two  of  three  principal  axes  be  equal, 
the  moments  of  all  axes  through  the  same  point  and  in  their 
plane  will  also  be  equal ;  and  if  the  moments  of  the  three 
principal  axes  through  a  point  be  equal,  the  moments  of  all 
axes  whatever,  through  the  same  point,  will  be  equal. 

(192.)  If  the  moments  of  the  principal  axes  through  the 
centre  of  gravity  be  known,  the  moments  for  all  other  axes 
through  that  point  may  be  easily  computed.  To  effect  this 
it  is  only  necessary  to  multiply  the  moments  of  the  principal 
axes  by  the  squares  of  the  co-sines  of  the  angles  formed  by 
them  respectively  with  the  axis  whose  moment  is  sought. 
The  products,  being  added  together,  will  give  the  required 
moment. 


THE    ELEMENTS    OF    MECHANICS.  CHAP.    X. 

(193.)  By  combining  this  result  with  that  of  (189.),  it  will 
be  evident  that  the  moment  of  all  axes  whatever  may  be  de- 
termined, if  those  of  the  principal  axes  through  the  centre 
of  gravity  be  known. 

(194.)  It  is  obvious  that  the  principal  axis  of  least  moment 
through  the  centre  of  gravity  has  a  less  moment  of  inertia 
than  any  other  axis  whatever.  For  it  has,  by  its  definition 
(190.),  a  less  moment  of  inertia  than  any  other  axis  through 
Ihe  centre  of  gravity,  and  every  other  axis  through  the  cen- 
tre of  gravity  has  a  less  moment  of  inertia  than  a  parallel 
axis  through  any  other  point  (1H7.)  and  (189.). 

(195.)  If  two  of  the  principal  axes  through  the  centre  of 
gravity  have  equal  moments  of  inertia,  all  axes  in  any  plane 
parallel  to  the  plane  of  these  axes,  and  passing  through  the 
point  where  a  perpendicular  from  the  centre  of  gravity  meets 
that  plane,  must  have  equal  moments  of  inertia.  For  by 
(191.)  all  axes  in  the  plane  of  those  two  have  equal  moments, 
and  by  (189.)  the  axes  in  the  parallel  plane  have  moments 
which  exceed  these  by  the  same  quantity,  being  equally  dis- 
tant from  them.  (187.) 

Hence  it  is  obvious  that  if  the  three  principal  axes  through 
the  centre  of  gravity  have  equal  moments,  all  axes  situated 
in  any  given  plane,  and  passing  through  the  point  where  the 
perpendicular  from  the  centre  of  gravity  meets  that  plane, 
will  have  equal  moments,  being  equally  distant  from  parallel 
axes  through  the  centre  of  gravity. 

(190.)  If  the  three  principal  axes  through  the  centre  of 
gravity  have  unequal  moments,  there  is  no  point  whatever  for 
which  all  axes  will  have  equal  moments  ;  but  if  the  principal 
axis  of  least  moment  and  the  intermediate  principal  axis 
through  the  centre  of  gravity  have  equal  moments,  then 
there  will  be  two  points  on  the  principal  axis  of  greatest  mo- 
ment, equally  distant  at  opposite  sides  of  the  centre  of  gravity, 
at  which  all  axes  will  have  equal  moments.  If  the  three 
principal  axes  through  the  centre  of  gravity  have  equal  mo- 
ments, no  other  point  of  the  body  can  have  principal  axes  of 
equal  moment. 

(197.)  When  a  body  revolves  on  a  fixed  axis,  the  parts  of 
its  mass  are  whirled  in  circles  round  the  axis ;  and  since  they 
move  with  a  common  angular  velocity,  they  will  have  centrifu- 
gal forces  proportional  to  their  distances  from  the  axis.  If 
the  component  parts  of  the  mass  were  not  united  together  by 
cohesive  forces  of  energies  greater  than  these  centrifugal 


CHAP.  X.  PRESSURE    UPON    AN    AXIS.  119 

forces,  they  would  be  separated,  and  would  fly  off  from  the 
axis ;  but  their  cohesion  prevents  this,  and  causes  the  effects 
of  the  different  centrifugal  forces,  which  affect  the  different 
parts  of  the  mass,  to  be  transmitted  so  as  to  modify  each  other, 
and  finally  to  produce  one  or  more  forces  mechanically  equiva- 
lent to  the  whole,  and  which  are  exerted  upon  the  axis  and 
resisted  by  it.  We  propose  now  to  explain  these  effects,  as 
far  as  it  is  possible  to  render  them  intelligible  without  the  aid 
of  mathematical  language. 

It  is  obvious  that  any  number  of  equal  parts  of  the  mass, 
which  are  uniformly  arranged  in  a  circle  round  the  axis,  have 
equal  centrifugal  forces  acting  from  the  centre  of  the  circle 
in  every  direction.  These  mutually  neutralize  each  other, 
and  therefore  exert  no  force  on  the  axis.  The  same  may  be 
said  of  all  parts  of  the  mass  which  are  regularly  and  equally 
distributed  on  every  side  of  the  axis. 

Also,  if  equal  masses  be  placed  at  equal  distances  on  oppo- 
site sides  of  the  axis,  their  centrifugal  forces  will  destroy  each 
other.  Hence  it  appears  that  the  pressure  which  the  axis  of 
rotation  sustains  from  the  centrifugal  forces  of  the  revolving 
mass,  arises  from  the  unequal  distribution  of  the  matter 
around  it. 

From  this  reasoning  it  will  be  easily  perceived  that,  in  the 
following  examples,  the  axis  of  rotation  will  sustain  no  pres- 
sure. 

A  globe  revolving  on  any  of  its  diameters,  the  density  be- 
ing the  same  at  equal  distances  from  the  centre. 

A  spheroid  or  a  cylinder  revolving  on  its  axis,  the  density 
being  equal  at  equal  distances  from  the  axis. 

A  cube  revolving  on  an  axis  which  passes  through  the  cen- 
tre of  two  opposite  bases,  being  of  uniform  density. 

A  circular  plate  of  uniform  thickness  and  density  revolving 
on  one  of  its  diameters  as  an  axis. 

(198.)  In  all  these  examples,  it  will  be  observed  that  the 
axis  of  rotation  passes  through  the  centre  of  gravity.  The 
general  theorem,  of  which  they  are  only  particular  instances, 
is,  "  If  a  body  revolve  on  a  principal  axis,  passing  through 
the  centre  of  gravity,  the  axis  will  sustain  no  pressure  from 
the  centrifugal  force  of  the  revolving  mass."  This  is  a  prop- 
erty in  which  the  principal  axes  through  the  centre  of  gravity 
are  unique.  There  is  no  other  axis  on  which  a  body  could 
revolve  without  pressure. 

If  two  of  the  principal  axes  through  the  centre  of  gravity 


120  THE  -ELEMENTS  OF  MECHANICS.  CHAP.  x. 

have  equal  moments,  every  axis  in  their  plane  has  the  same 
moment,  and  is  to  be  considered  equally  as  a  principal  axis. 
In  this  case,  the  body  would  revolve  on  any  of  these  axes  with- 
out pressure. 

A  homogeneous  spheroid  furnishes  an  example  of  this.  If 
any  of  the  diameters  of  the  earth's  equator  were  a  fixed  axis, 
the  earth  would  revolve  on  it  without  producing  pressure. 

If  the  three  principal  axes  through  the  centre  of  gravity 
have  equal  moments,  all  axes  through  the  centre  of  gravity 
are  to  be  considered  as  principal  axes.  In  this  case,  the  body 
would  revolve  without  pressure  on  any  axis  through  the  cen- 
tre of  gravity. 

A  globe,  iii  which  the  density  of  the  mass  at  equal  distances 
from  the  centre  is  the  same,  is  an  example  of  this.  Such  a 
body  would  revolve  without  pressure  on  any  axis  through  its 
centre. 

(199.)  Since  no  pressure  is  excited  on  the  axis  in  these 
cases,  the  state  of  the  body  will  not  be  changed,  if,  during  its 
rotation,  the  axis  cease  to  be  fixed.  The  body  will,  notwith- 
standing, continue  to  revolve  round  the  axis,  and  the  axis  will 
maintain  its  position. 

Thus  a  spinning-top  of  homogeneous  material  and  sym- 
metrical form  will  revolve  steadily  in  the  same  position,  until 
the  friction  of  its  point  with  the  surface  on  which  it  rests  de- 
prives it  of  motion.  This  is  a  phenomenon  which  can  only 
be  exhibited  when  the  axis  of  rotation  is  a  principal  axis 
through  the  centre  of  gravity. 

(200.)  If  the  body  revolve  round  any  axis  through  the  cen- 
tre of  gravity,  which  is  not  a  principal  axis,  the  centrifugal 
pressure  is  represented  by  two  forces,  which  are  equal  and 
parallel,  but  which  act  in  opposite  directions  on  different  points 
of  the  axis.  The  effect  of  these  forces  is  to  produce  a  strain 
upon  the  axis,  and  give  the  body  a  tendency  to  move  round 
another  axis  at  right  angles  to  the  former. 

.  (201.)  If  the  fixed  axis  on  which  a  body  revolves  be  a 
principal  axis  through  any  point  different  from  the  centre  of 
gravity,  then  a  pressure  will  be  produced  by  the  centrifugal 
force  of  the  revolving  mass,  and  this  pressure  will  act  at  right 
angles  to  the  axis  on  the  point  to  which  it  is  a  principal  axis, 
and  in  the  plane  through  that  axis  and  the  centre  of  gravity. 
The  amount  of  the  pressure  will  be  proportional  to  the  mass 
of  the  body,  the  distance  of  the  centre  of  gravity  from  the 
axis,  and  the  square  of  the  velocity  of  rotation. 


CHAP.  X.        PRESSURE  UPON  AN  AXIS.  121 

(1202.)  Since  the  whole  pressure  is  in  this  case  excited  on 
a  single  point,  the  stability  of  the  axis  will  not  be  disturb- 
ed, provided  that  point  alone  be  fixed.  So  that,  even  though 
the  axis  should  be  free  to  turn  on  that  point,  no  motion  will 
ensue  as  long  as  no  external  forces  act  upon  the  body. 

(203.)  If  the  axis  of  rotation  be  not  a  principal  axis,  the 
centrifugal  forces  will  produce  an  effect  which  cannot  be  rep- 
resented by  a  single  force.  The  effect  may  be  understood  by 
conceiving  two  forces  to  act  on  different  points  of  the  axis  at 
right  angles  to  it  and  to  each  other.  The  quantities  of  these 
pressures  and  their  directions  depend  on  the  figure  and  density 
of  the  mass  and  the  position  of  the  axis,  in  a  manner  which 
cannot  be  explained  without  the  aid  of  mathematical  language 
and  principles. 

(204.)  The  effects  upon  the  axis  which  have  been  now  ex- 
plained are  those  which  arise  from  the  motion  of  rotation, 
from  whatever  cause  that  motion  may  have  arisen.  The 
forces  which  produce  that  motion,  however,  are  attended  with 
effects  on  the  axis  which  still  remain  to  be  noticed.  When 
these  forces,  whether  they  be  of  the  nature  of  instantaneous 
actions  or  continued  forces,  are  entirely  resisted  by  the  axis, 
their  directions  must  severally  be  in  a  plane  passing  through 
the  axis,  or  they  must,  by  the  principles  of  the  composition 
of  force  [(74.)  et  seq.],  be  mechanically  equivalent  to  forces 
in  that  plane.  In  every  other  case,  the  impressed  forces  must 
produce  motion,  and,  except  in  certain  cases,  must  also  pro- 
duce effects  upon  the  axis. 

By  the  rules  for  the  composition  of  force  it  is  possible  in  all 
cases  to  resolve  the  impressed  forces  into  others  which  are 
either  in  planes  through  the  axis,  or  in  planes  perpendicular 
to  it,  or,  finally,  some  in  planes  through  it,  and  others  in 
planes  perpendicular  to  it.  The  effect  of  those  which  are  in 
planes  through  the  axis  has  been  already  explained ;  and  we 
shall  now  confine  our  attention  to  those  impelling  forces 
which  act  at  right  angles  to  the  axis,  and  which  produce  motion. 

It  will  be  sufficient  to  consider  the  effect  of  a  single  force 
at  right  angles  to  the  axis ;  for  whatever  be  the  number  of 
forces  which  act  either  simultaneously  or  successively,  the 
effect  of  the  whole  will  be  decided  by  combining  their  sepa- 
rate effects,  The  effect  which  a  single  force  produces,  de- 
pends on  two  circumstances — 1.  The  position  of  the  axis  with 
respect  to  the  figure  and  mass  of  the  body,  and,  2.  The  quan- 
tity and  direction  of  the  force  itself. 
11 


122  TIIE    ELEMENTS    OF    MECHANICS.  CHAP.  X. 

In  general,  the  shock  which  the  axis  sustains  from  the  im- 
pact may  be  represented  by  two  impacts  applied  to  it  at  differ- 
ent points,  one  parallel  to  the  impressed  force,  and  the  other 
perpendicular  to  it,  but  both  perpendicular  to  the  axis.  There 
are  certain  circumstances,  however,  under  which  this  effect 
will  be  modified. 

If  the  impulse  which  the  body  receives  be  in  a  direction 
perpendicular  to  a  plane  through  the  axis  and  the  centre  of 
gravity,  and  at  a  distance  from  the  axis  which  bears  to  the 
radius  of  gyration  (186.)  the  same  proportion  as  that  line  bears 
to  the  distance  of  the  centre  of  gravity  from  the  axis,  there 
are  certain  cases  in  which  the  impulse  will  produce  no  per- 
cussion. To  characterize  these  cases  generally  would  require 
analytical  formula  which  cannot  conveniently  be  translated 
into  ordinary  language.  That  point  of  the  plane,  however, 
where  the  direction  of  the  impressed  force  meets  it,  when  no 
percussion  on  the  axis  is  produced,  is  called  the  centre  of 
percussion. 

If  the  axis  of  rotation  be  a  principal  axis,  the  centre  of  per- 
cussion must  be  in  the  right  line  drawn  through  the  centre  of 
gravity,  intersecting  the  axis  at  right  angles,  and  at  the  dis- 
tance from  the  axis  already  explained. 

If  the  axis  of  rotation  be  parallel  to  a  principal  axis  through 
the  centre  of  gravity,  the  centre  of  percussion  will  be  deter- 
mined in  the  same  manner. 

(205.)  There  are  many  positions  which  the  axis  may  have, 
in  which  there  will  be  no  centre  of  percussion ;  that  is,  there 
will  be  no  direction  in  which  an  impulse  could  be  applied 
without  producing  a  shock  upon  the  axis.  One  of  these 
positions  is  when  it  is  a  principal  axis  through  the  centre  of 
gravity.  This  is  the  only  case  of  rotation  round  an  axis,  in 
which  no  effect  arises  from  the  centrifugal  force ;  and  there- 
fore it  follows  that  the  only  case  in  which  the  axis  sustains  no 
effect  from  the  motion  produced,  is  one  in  which  it  must 
necessarily  suffer  an  effect  from  that  which  produces  the  mo- 
tion. 

If  the  body  be  acted  upon  by  continued  forces,  their  effect 
is  at  each  instant  determined  by  the  general  principles  for  the 
composition  of  force. 


CHAP.    XI  THE    PENDULUM.  123 

CHAPTER  XI. 

ON    THE    PENDULUM. 

(206.)  WHEN  a  body  is  placed  on  an  horizontal  axis  which 
does  not  pass  through  its  centre  of  gravity,  it  will  remain  in 
permanent  equilibrium  only  when  the  centre  of  gravity  is  im- 
mediately below  the  axis.  If  this  point  be  placed  in  any  other 
situation,  the  body  will  oscillate  from  side  to  side,  until  the 
atmospherical  resistance  and  the  friction  of  the  axis  destroy 
its  motion.  (159,160.)  Such  a  body  is  called  a  pendulum. 
The  swinging  motion  which  it  receives  is  called  oscillation  or 
vibration. 

(207.)  The  use  of  the  pendulum,  not  only  for  philosophi- 
cal purposes,  but  in  the  ordinary  economy  of  life,  renders  it 
a  subject  of  considerable  importance.  It  furnishes  the  most 
exact  means  of  measuring  time,  and  of  determining  with 
precision  various  natural  phenomena.  By  its  means  the  vari- 
ation of  the  force  of  gravity  in  different  latitudes  is  discover- 
ed, and  the  law  of  that  variation  experimentally  exhibited. 
In  the  present  chapter,  we  propose  to  explain  the  general 
principles  which  regulate  the  oscillation  of  pendulums.  Mi- 
nute details  concerning  their  construction  will  be  given  in  the 
twenty-first  chapter  of  this  volume. 

(208.)  A  simple  pendulum  is  composed  of  a  heavy  molecule 
attached  to  the  end  of  a  flexible  thread,  and  suspended  by  a 
fixed  point  O,  Jig.  73.  When  the  pendulum  is  placed  in  the 
position  O  C,  the  molecule  being  vertically  below  the  point 
of  suspension,  it  will  remain  in  equilibrium  ;  but  if  it  be  drawn 
into  the  position  O  A,  and  there  liberated,  it  will  descend 
towards  C,  moving  through  the  arc  A  C  with  accelerated  mo- 
tion. Having  arrived  at  C,  and  acquired  a  certain  velocity,  it 
will,  by  reason  of  its  inertia,  continue  to  move  in  the  same 
direction.  It  will  therefore  commence  to  ascend  the  arc  C  A' 
with  the  velocity  so  acquired.  During  its  ascent,  the  weight 
of  the  molecule  retards  its  motion  in  exactly  the  same  manner 
as  it  had  accelerated  it  in  descending  from  A  to  C  ;  and  when 
the  molecule  has  ascended  through  the  arc  C  A'  equal  to  C  A, 
its  entire  velocity  will  be  destroyed,  and  it  will  cease  to  move 
in  that  direction.  It  will  thus  be  placed  at  A'  in  the  same 
manner  as  in 'the  first  instance  it  had  been  placed  at  A,  and 


124  THE    ELEMENTS    OP    MECHANICS.  CHAP.    XI 

consequently  it  will  descend  from  A'  to  C  with  accelerated 
motion,  in  the  same  manner  as  it  first  moved  from  A  to  C.  It 
will  then  ascend  from  C  to  A,  and  so  on  continually.  In 
this  case,  the  thread,  by  which  the  molecule  is  suspended,  is 
supposed  to  be  perfectly  flexible,  inextensible,  and  of  incon- 
siderable weight.  The  point  of  suspension  is  supposed  to  be 
without  friction,  and  the  atmosphere  to  offer  no  resistance  to 
the.  motion. 

It  is  evident  from  what  has  been  stated,  that  the  times  of 
moving  from  A  to  A'  and  from  A'  to  A  are  equal,  and  will 
continue  to  be  equal  so  long  as  the  pendulum  continues  to 
vibrate.  If  the  number  of  vibrations  performed  by  the  pen- 
dulum were  registered,  and  the  time  of  each  vibration  known, 
this  instrument  would  become  a  chronometer. 

The  rate  at  which  the  motion  of  the  pendulum  is  accele- 
rated in  its  descent  towards  its  lowest  position  is  not  uniform, 
because  the  force  which  impels  it  is  continually  decreasing, 
and  altogether  disappears  at  the  point  C.  The  impelling  force 
arises  from  the  effect  of  gravity  on  the  suspended  molecule, 
and  this  effect  is  always  produced  in  the  vertical  direction  A 
V.  The  greater  the  angle  O  A  V  is,  the  less  efficient  the 
force  of  gravity  will  be  in  accelerating  the  molecule :  this 
angle  evidently  increases  as  the  molecule  approaches  C,  which 
will  appear  by  inspecting  jig.-  73.  At  C,  the  force  of  gravity 
acting  in  the  direction  C-B  is  totally  expended  in  giving  ten- 
sion to  the  thread,  and  is>  inefficient  in  moving  the  molecule. 
It  follows,  therefore,  thai  the  impelling  force  is  greatest  at  A, 
and  continually  diminishes  from  A  to  C,  where  it  altogether 
vanishes.  The  same  observations  will  be  applicable  to  the  re- 
tarding force  from  C  to  A',  and  to  the  accelerating  force  from 
A'  to  C,  and  so  on. 

When  the  length  of  the  thread  and  the  intensity  of  the 
force  of  gravity  are  given,  the  time  of  vibration  depends  on 
the  length  of  the  arc  A  C,  or  on  the  magnitude  of  the  angle 
A  O  C.  If,  however,  this  angle  do  not  exceed  a  certain 
limit  of  magnitude,  the  time  of  vibration  will  be  subject  to 
no  sensible  variation,  however  that  angle  may  vary.  Thus 
the  time  of  oscillation  will  be  the  same,  whether  the  angle 
A  O  C  be  2°,  or  1°  30',  or  1°,  or  any  lesser  magnitude.  This 
property  of  a  pendulum  is  expressed  by  the  word  isochronism. 
The  strict  demonstration  of  this  property  depends  on  math- 
ematical principles,  the  details  of  which  would  not  be  suita- 


CHAP.  XI.  THE  PENDULUM.  125 

ble  to  the  present  treatise.  It  is  not  difficult,  however,  to 
explain  generally  how  it  happens  that  the  same  pendulum 
will  swing  through  greater  and  smaller  arcs  of  vibration  in 
the  same  time.  If  it  swing  from  A,  the  force  of  gravity  at 
the  commencement  of  its  motion  impels  it  with  an  effect 
depending  on  the  obliquity  of  the  lines  O  A  and  A  V.  If  it 
commence  its  motion  from  a,  the  impelling  effect  from  the 
force  of  gravity  will  be  considerably  less  than  at  A  ;  conse- 
quently, the  pendulum  begins  to  move  at  a  slower  rate,  when 
it  swings  from  a  than  when  it  moves  from  A  :  the  greater 
magnitude  of  the  swing  is  therefore  compensated  by  the  in- 
creased velocity,  so  that  the  greater  and  the  smaller  arcs  of 
vibration  are  moved  through  in  the  same  time. 

(209.)  To  establish  this  property  experimentally,  it  is  only 
necessary  to  suspend  a  small  ball  of  metal,  or  other  heavy 
substance,  by  a  flexible  thread,  and  to  put  it  in  a  state  of 
vibration,  the  entire  arc  of  vibration  not  exceeding  4°  or  5°, 
the  friction -on  the  point  of  suspension  and  other  causes  will 
gradually  diminish  the  arc  of  vibration,  so  that,  after  the  lapse 
of  some  hours,  it  will  be  so  small,  that  the  motion  will  scarcely 
be  discerned  without  microscopic  aid.  If  the  vibration  of 
this  pendulum  be  observed  in  reference  to  a  correct  time- 
keeper, at  the  commencement,  at  the  middle,  and  towards 
the  end  of  its  motion,  the  rate  will  be  found  to  suffer  no 
sensible  change. 

This  remarkable  law  of  isochronism  was  one  of  the  earliest 
discoveries  of  Galileo.  It  is  said,  that,  when  very  young,  he 
observed  a  chandelier  suspended  from  the  roof  of  a  church 
in  Pisa  swinging  with  a  pendulous  motion,  and  was  struck 
with  the  uniformity  of  the  rate,  even  when  the  extent  of  the 
swing  was  subject  to  evident  variation. 

(•210.)  It  has  been  stated  in  (117.)  that  the  attraction  of 
gravity  affects  all  bodies  equally,  and  moves  them  with  the 
same  velocity,  whatever  be  the  nature  or  quantity  of  the  ma- 
terials of  which  they  are  composed.  Since  it  is  the  force 
of  gravity  which  moves  the  pendulum,  we  should  therefore 
expect  that  the  circumstances  of  that  motion  should  not  be 
affected  either  by  the  quantity  or  quality  of  the  pendulous 
body.  And  we  find  this,  in  fact,  to  be  the  case  ;  for  if  small 
pieces  of  different  heavy  substances,  such  as  lead,  brass,  ivory, 
&,c.,  be  suspended  by  fine  threads  of  equal  length,  they  will 
vibrate  in  the  same  time,  provided  their  weights  bear  a  con- 
11* 


126  THE  ELEMENTS  OF  MECHANICS.      CHAP.  XI 

siderable  proportion  to  the  atmospherical  resistance,  or  that 
they  be  suspended  in  vacuo. 

(211.)  Since  the  time  of  vibration  of  a  pendulum,  which 
oscillates  in  small  arcs,  depends  neither  on  the  magnitude 
of  the  arc  of  vibration  nor  on  the  quality  or  weight  of  the 
pendulous  body,  it  will  be  necessary  to  explain  the  circum- 
stances on  which  the  variation  of  this  time  depends. 

The  first  and  most  striking  of  these  circumstances  is  the 
length  of  the  suspending  thread.  The  rudest  experiments 
will  demonstrate  the  fact,  that  every  increase  in  the  length  of 
this  thread  will  produce  a  corresponding  increase  in  the  time 
of  vibration  ;  but  according  to  what  law  does  this  increase  pro- 
ceed ?  If  the  length  of  the  thread  be  doubled  or  trebled,  will 
the  time  of  vibration  also  be  increased  in  a  double  or  treble 
proportion  ?  This  problem  is  capable  of  exact  mathematical 
solution,  and  the  result  shows  that  the  time  of  vibration  in- 
creases, not  in  the  proportion  of  the  increased  length  of  the 
thread,  but  as  the  square  root  of  that  length  ;  that  is  to  say, 
if  the  length  of  the  thread  be  increased  in  a  four-fold  propor- 
tion, the  time  of  vibration  will  be  augmented  in  a  two-fold 
proportion.  If  the  thread  be  increased  to  nine  times  its 
length,  the  time  of  vibration  will  be  trebled,  and  so  on.  This 
relation  is  exactly  the  same  as  that  which  was  proved  to  sub- 
sist between  the  spaces  through  which  a  body  falls  freely, 
and  the  times  of  fall.  In  the  table,  page  75,  if  the  figures 
representing  the  height  be  understood  to  express  the  length 
of  different  pendulums,  the  figures  immediately  above  them 
will  express  the  corresponding  times  of  vibration. 

This  law  of  the  proportion  of  the  lengths  of  pendulums  to 
the  squares  of  the  time  of  vibration  may  be  experimentally 
established  in  the  following  manner  : — 

Let  A,  B,  C,  Jig.  74.,  be  three  small  pieces  of  metal,  each 
attached  by  threads  to  two  points  of  suspension,  and  let  them 
be  placed  in  the  same  vertical  line  under  the  point  O  ;  sup- 
pose them  so  adjusted  that  the  distances  O  A,  OB,  and  O  C, 
.shall  be  in  the  proportion  of  the  numbers  1,  4,  and  9.  Let 
them  be  removed  from  the  vertical  in  a  direction  at  right 
angles  to  the  plane  of  the  paper,  so  that  the  threads  shall  be 
in  the  same  plane,  and  therefore  the  three  pendulums  will 
have  the  same  angle  of  vibration.  Being  now  liberated,  the 
pendulum  A  will  immediately  gain  upon  B,  and  B  upon  C,  so 
that  A  will  have  completed  one  vibration  before  Bor  C.  At  the 


CHAP.  XI.  THE  PENDULUM.  127 

end  of  the  second  vibration  of  A,  the  pendulum  B  will  have 
arrived  at  the  end  of  its  first  vibration,  so  that  the  suspend- 
ing threads  of  A  and  B  will  then  be  separated  by  the  whole 
angle  of  vibration  ;'  at  the  end  of  the  fourth  vibration  of  A, 
the  suspending  threads  of  A  and  B  will  return  to  their  first 
position,  B  having  completed  two  vibrations;  thus  the  propor- 
tion of  the  times  of  vibration  of  B  and  A  will  be  2  to  1,  the 
proportion  of  their  lengths  being  4  to  1.  At  the  end  of  the 
third  vibration  of  A,  C  will  have  completed  one  vibration, 
and  the  suspending  strings  will  coincide  in  the  position  dis- 
tant by  the  whole  angle  of  vibration  from  their  first  position. 
So  that  three  vibrations  of  A  are  performed  in  the  same  time 
as  one  of  C  :  the  proportion  of  the  time  of  vibration  of  C  and 
A  is,  therefore,  3  to  1,  the  proportion  of  their  lengths  being 
9  to  1,  conformably  to  the  law  already  explained. 

(  1:2.)  [n  all  the  preceding  observations  we  have  assumed 
that  the  material  of  the  pendulous  body  is  of  inconsiderable 
magnitude,  its  whole  weight  being  conceived  to  be  collected 
into  a  physical  point.  This  is  generally  called  a  simple  pen- 
dulum ;  but  since  the  conditions  of  a  suspending  thread  without 
weight,  and  a  heavy  molecule  without  magnitude,  cannot  have 
practical  existence,  the  simple  pendulum  must  be  considered 
as  imaginary,  and  merely  used  to  establish  hypothetical  theo- 
rems, which,  though  inapplicable  in  practice,  are  nevertheless 
the  means  of  investigating  the  laws  which  govern  the  real 
phenomena  of  pendulous  bodies. 

A  pendulous  body  being  of  determinate  magnitude,  its 
several  parts  will  be  situated  at  different  distances  from  the 
axis  of  suspension.  If  each  component  part  of  such  a 
body  were  separately  connected  with  the  axis  of  suspension 
by  a  fine  thread,  it  would,  if  unconnected  with  the  other 
particles,  be  an  independent  simple  pendulum,  and  would 
oscillate  according  to  the  laws  already  explained.  It  there- 
fore follows  that  those  particles  of  the  body  which  are  nearest 
to  the  axis  of  suspension  would,  if  liberated  from  their  con- 
nection with  the  others,  vibrate  more  rapidly  than  those  which 
are  more  remote.  The  connection,  however,  which  the  par- 
ticles of  the  body  have,  by  reason  of  their  solidity,  compels 
them  all  to  vibrate  in  the  same  time.  Consequently,  those 
particles  which  are  nearest  the  axis  are  retarded  by  the  slower 
motion  of  those  which  are  more  remote  ;  while  the  more 
remote  particles,  on  the  other  hand,  are  urged  forward  by 
the  greater  tendency  of  the  nearer  particles  to  rapid  vibration 


128  THE  ELEMENTS  OF  MECHANICS.       CHAP.  XI. 

This  will  be  more  readily  comprehended,  if  we  conceive  two 
particles  of  matter,  A  and  B,  Jiff.  75.,  to  be  connected  with 
the  same  axis  O  by  an  inflexible  wire  O  C,  the  weight  of 
which  may  be  neglected.  If  B  were  removed,  A  would  vi- 
brate in  a  certain  time  depending  upon  the  distance  O  A. 
If  A  were  removed,  and  B  placed  upon  the  wire  at  a  distance 
B  O  equal  to  four  times  A  O,  B  would  vibrate  in  twice  the 
former  time.  Now,  if  both  be  placed  on  the  wire  at  the 
distances  just  mentioned,  the  tendency  of  A  to  vibrate  more 
rapidly  will  be  transmitted  to  B  by  means  of  the  wire,  and 
will  urge  B  forward  more  quickly  than  if  A  were  not  present: 
on  the  other  hand,  the  tendency  of  B  to  vibrate  more  slowly 
will  be  transmitted  by  the  wire  to  A,  and  will  cause  it  to 
move  more  slowly  than  if  B  were  not  present.  The  inflex- 
ible quality  of  the  connecting  wire  will  in  this  case  compel 
A  and  B  to  vibrate  simultaneously,  the  time  of  vibration  be- 
ing greater  than  that  of  A,  and  less  than  that  of  B,  if  each 
vibrated  unconnected  with  the  other. 

If,  instead  of  supposing  two  particles  of  matter  placed  on 
the  wire,  a  greater  number  were  supposed  to  be  placed  at 
various  distances  from  O,  it  is  evident  the  same  reasoning 
would  be  applicable.  They  would  mutually  affect  each  other's 
motion  ;  those  placed  nearest  to  point  O  accelerating  the 
motion  of  those  more  remote,  and  being  themselves  retarded 
by  the  latter.  Among  these  particles  one  would  be  found  in 
which  all  these  effects  would  be  mutually  neutralized,  all  the 
particles  nearer  O  being  retarded  in  reference  to  that  motion 
which  they  would  have  if  unconnected  with  the  rest,  and 
those  more  remote  being  in  the  same  respect  accelerated. 
The  point  at  which  such  a  particle  is  placed  is  called  the 
centre  of  oscillation. 

What  has  been  here  observed  of  the  effects  of  particles 
of  matter  placed  upon  rigid  wire  will  be  equally  applicable  to 
the  particles  of  a  solid  body.  Those  which  are  nearer  to 
the  axis  are  urged  forward  by  those  which  are  more  remote, 
and  are,  in  their  turn,  retarded  by  them ;  and,  as  with  the 
particles  placed  upon  the  wire,  there  is  a  certain  particle 
of  the  body  at  which  the  effects  are  mutually  neutralized, 
and  which  vibrates  in  the  same  time  as  it  would  if  it  were 
unconnected  with  the  other  parts  of  the  body,  and  simply 
connected  by  a  fine  thread  to  the  axis.  By  this  centre  of 
oscillation  the  calculations  respecting  the  vibration  of  a  solid 
body  are  rendered  as  simple  as  those  of  a  molecule  of  incon- 


CHAP.  XI.  CENTRE  OF  OSCILLATION.  129 

siderable  magnitude.  All  the  properties  which  have  been 
explained  as  belonging  to  a  simple  pendulum  may  thus  be 
transferred  to  a  vibrating  body  of  any  magnitude  and  figure, 
by  considering  it  as  equivalent  to  a  single  particle  of  matter 
vibrating  at  its  centre  of  oscillation. 

(213.)  It  follows  from  this  reasoning,  that  the  virtual  length 
of  a  pendulum  is  to  be  estimated  by  the  distance  of  its  centre 
of  oscillation  from  the  axis  of  suspension,  and  therefore  that 
the  times  of  .vibration  of  different  pendulums  are  in  the  same 
proportion  as  the  square  roots  of  the  distances  of  their  centres 
of  oscillation  from  their  axes. 

The  investigation  of  the  position  of  the  centre  of  oscilla- 
tion is,  in  most  cases,  a  subject  of  intricate  mathematical 
calculation.  It  depends  on  the  magnitude  and  figure  of  the 
pendulous  body,  the  manner  in  which  the  mass  is  distributed 
through  its  volume,  or  the  density  of  its  several  parts,  and 
the  position  of  the  axis  on  which  it  swings. 

The  place  of  the  centre  of  oscillation  may  be  determined 
when  the  position  of  the  centre  of  gravity  and  the  centre  of 
gyration  are  known  ;  for  the  distance  of  the  centre  of  oscilla- 
tion from  the  axis  will  always  be  obtained  by  dividing  the 
square  of  the  radius  of  gyration  (180.)  by  the  distance  of  the 
centre  of  gravity  from  the  axis.  Thus,  if  6  be  the  radius 
of  gyration,  and  9  the  distance  of  gravity  from  the  axis,  36 
divided  by  9,  which  is  4,  will  be  the  distance  of  the  centre 
of  oscillation  from  the  axis.  Hence  it  may  be  inferred  gen 
erally,  that  the  greater  the  proportion  which  the  radius  of 
gyration  bears  to  the  distance  of  the  centre  of  gravity  from 
the  axis,  the  greater  will  be  the  distance  of  the  centre  of  os- 
cillation. 

It  follows  from  this  reasoning,  that  the  length  of  a  pen- 
dulum is  not  limited  by  the  dimensions  of  its  volume.  If  the 
axis  be  so  placed  that  the  centre  of  gravity  is  near  it,  and 
the  centre  of  gyration  comparatively  removed  from  it,  the 
centre  of  oscillation  may  be  placed  far  beyond  the  limits  of 
the  pendulous  body.  Suppose  the  centre  of  gravity  is  at  a 
distance  of  one  inch  from  the  axis,  and  the  centre  of  gyra- 
tion 12  inches,  the  centre  of  oscillation  will  then  be  at  the 
distance  of  144  inches,  or  12  feet.  Such  a  pendulum  may 
not,  in  its  greatest  dimensions,  exceed  one  foot,  and  yet  its 
time  of  vibration  would  be  equal  to  that  of  a  simple  pendulum 
whose  length  is  12  feet. 

By  these  means  pendulums  of  small  dimensions  may  be 


130  T^IE  ELEMENTS  OF  MECHANICS.  CHAP.  XI. 

made  to  vibrate  as  slowly  as  may  be  desired.  The  instru- 
ments called  metronomes,  used  for  marking  the  time  of  musical 
performances,  are  constructed  on  this  principle. 

(214.)  The  centre  of  oscillation  is  distinguished  by  a  very 
remarkable  property  in  relation  to  the  axis  of  suspension.  If 
A,  Jig-  76.,  be  the  point  of  suspension,  and  O  the  correspond- 
ing centre  of  oscillation,  the  time  of  vibration  of  the  pendu- 
lum will  not  be  changed  if  it  be  raised  from  its  support, 
inverted,  and  suspended  from  the  point  O.  It  follows,  there- 
fore, that  if  O  be  taken  as  the  point  of  suspension,  A  will 
be  the  corresponding  centre  of  oscillation.  These  two  points 
are,  therefore,  convertible.  This  property  may  be  verified 
experimentally  in  the  following  manner.  A  pendulum  being 
put  into  a  state  of  vibration,  let  a  small  heavy  body  be  sus- 
pended by  a  fine  thread,  the  length  of  which  is  so  adjusted 
that  it  vibrates  simultaneously  with  the  pendulum.  Let  the 
distance  from  the  point  of  suspension  to  the  centre  of  the 
vibrating  body  be  measured,  and  take  this  distance  on  the 
pendulum  from  the  axis  of  suspension  downwards  ;  the  place 
of  the  centre  of  oscillation  will  thus  be  obtained,  since  the 
distance  so  measured  from  the  axis  is  the  length  of  the  equiv- 
alent simple  pendulum.  If  the  pendulum  be  now  raised 
from  its  support,  inverted,  and  suspended  from  the  centre 
of  oscillation  thus  obtained,  it  will  be  found  to  vibrate  simul- 
taneously with  the  body  suspended  by  the  thread. 

(215.)  This  property  of  the  interchangeable  nature  of  the 
centres  of  oscillation  and  suspension  has  been,  at  a  late 
period,  adopted  by  Captain  Kater,  as  an  accurate  means  of 
determining  the  length  of  a  pendulum.  Having  ascertained 
with  great  accuracy  two  points  of  suspension  at  which  the 
same  body  will  vibrate  in  the  same  time,  the  distance  be- 
tween these  points,  being  accurately  measured,  is  the  length 
of  the  equivalent  simple  pendulum.  Se»  Chapter  XXI. 

(216.)  The  manner  in  which  the  time  of  vibration  of  a 
pendulum  depends  on  its  length  being  explained,  we  are 
next  to  consider  how  this  time  is  affected  by  the  attraction 
of  gravity.  It  is  obvious  that,  since  the  pendulum  is  moved 
by  this  attraction,  the  rapidity  of  its  motion  will  be  increased, 
if  the  impelling  force  receives  any  augmentation ;  but  it  still 
is  to  be  decided,  in  what  exact  proportion  the  time  of  oscilla- 
tion will  be  diminished  by  any  proposed  increase  in  the  in- 
tensity of  the  earth's  attraction.  It  can  be  demonstrated 
mathematically,  that  the  time  of  one  vibration  of  a  pendulum 


CHAP*    XI.  TIME    OF    VIBRATION.  131 

has  the  same  proportion  to  the  time  of  falling  freely  in  the 
perpendicular  direction,  through  a  height  equal  to  half  the 
length  of  the  pendulum,  as  the  circumference  of  a  circle  has 
to  its  diameter.  Since,  therefore,  the  times  of  vibration  of 
pendulums  are  in  a  fixed  proportion  to  the  times  of  falling 
freely  through  spaces  equal  to  the  halves  of  their  lengths,  it 
follows  that  these  times  have  the  same  relation  to  the  force 
of  attraction  as  the  times  of  falling  freely  through  their 
lengths  have  to  that  force.  If  the  intensity  of  the  force  of 
gravity  were  increased  in  a  four-fold  proportion,  the  time  of 
falling  through  a  given  height  would  be  diminished  in  a  two- 
fold proportion  ;  if  the  intensity  were  increased  to  a  nine-fold 
proportion,  the  time  of  falling  through  a  given  space  would 
be  diminished  in  a  three-fold  proportion,  and  so  on  ;  the  rate 
of  diminution  of  the  time  being  always  as  the  square  root 
of  the  increased  force.  By  what  has  been  just  stated,  this 
law  will  also  be  applicable  to  the  vibration  of  pendulums. 
Any  increase  in  the  intensity  of  the  force  of  gravity  would 
cause  a  given  pendulum  to  vibrate  more  rapidly,  and  the  in- 
creased rapidity  of  the  vibration  would  be  in  the  same  pro- 
portion as  the  square  root  of  the  increased  intensity  of  the 
force  of  gravity. 

(217.)  The  laws  which  regulate  the  times  of  vibration  of 
pendulums  in  relation  to  one  another  being  well  understood, 
the  whole  theory  of  these  instruments  will  be  completed, 
when  the  method  of  ascertaining  the  actual  time  of  vibration 
of  any  pendulum,  in  reference  to  its  length,  has  been  explain- 
ed. In  such  an  investigation,  the  two  elements  to  be  deter- 
mined are,  1.  the  exact  time  of  a  single  vibration,  and,  2.  the 
exact  distance  of  the  centre  of  oscillation  from  the  point  of 
suspension. 

The  former  is  ascertained  by  putting  a  pendulum  in  motion 
in  the  presence  of  a  good  chronometer,  and  observing  pre- 
cisely the  number  of  oscillations  which  are  made  in  any  pro- 
posed number  of  hours.  The  entire  time  during  which  the 
pendulum  swings,  being  divided  by  the  number  of  oscillations 
made  during  that  time,  the  exact  time  of  one  oscillation  will 
be  obtained. 

The  distance  of  the  centre  of  oscillation  from  the  point 
of  suspension  may  be  rendered  a  matter  of  easy  calculation, 
by  giving  a  certain  uniform  figure  and  material  to  the  pendu- 
lous body. 

(218.)  The  time  of  vibration  of  one  pendulum  of  known 


' 
132  THE  ELEMENTS  OF  MECHANICS.      CHAP.  XI. 

length  being  thus  obtained,  we  shall  be  enabled  immediately 
to  solve  either  of  the  following  problems. 

"  To  find  the  length  of  a  pendulum  which  shall  vibrate  in 
a  given  time." 

"  To  find  the  time  of  vibration  of  a  pendulum  of  a  given 
length." 

The  former  is  solved  as  follows :  the  time  of  vibration  of 
the  known  pendulum  is  to  the  time  of  vibration  of  the  requir- 
ed pendulum  as  the  square  root  of  the  length  of  the  known 
pendulum  is  to  the  square  root  of  the  length  of  'the  required 
pendulum.  This  length  is  therefore  found  by  the  ordinary 
rules  of  arithmetic. 

The  latter  may  be  solved  as  follows :  the  length  of  the 
known  pendulum  is  to  the  length  of  the  proposed  pendulum, 
as  the  square  of  the  time  of  vibration  of  the  known  pendu- 
lum is  to  the  square  of  the  time  of  vibration  of  the  proposed 
pendulum.  The  latter  time  may  therefore  be  found  by  arith- 
metic. 

(219.)  Since  the  rate  of  a  pendulum  has  a  known  relation 
to  the  intensity  of  the  earth's  attraction,  we  are  enabled, 
by  this  instrument,  not  only  to  detect  certain  variations  in 
that  attraction  in  various  parts  of  the  earth,  but  also  to 
discover  the  actual  amount  of  the  attraction  at  any  given 
place. 

The  actual  amount  of  the  earth's  attraction  at  any  given 
place  is  estimated  by  the  height  through  which  a  body  would 
fall  freely  at  that  place  in  any  given  time,  as  in  one  second. 
To  determine  this,  let  the  length  of  a  pendulum  which  would 
vibrate  in  one  second  at  that  place  be  found.  As  the  circum- 
ference of  a  circle  is  to  its  diameter  (a  known  proportion),  so 
will  one  second  be  to  the  time  of  falling  through  a  height 
equal  to  half  the  length  of  this  pendulum.  This  time  is 
therefore  a  matter  of  arithmetical  calculation.  It  has  been 
proved  in  (120.),  that  the  heights,  through  which  a  body  falls 
freely,  are  in  the  same  proportion  as  the  squares  of  the  times ; 
from  whence  it  follows,  that  the  square  of  the  time  of  falling 
through  a  height  equal  to  half  the  length  of  the  pendulum  is 
to  one  second  as  half  the  length  of  that  pendulum  is  to  the 
height  through  which  a  body  would  fall  in  one  second.  This 
height,  therefore,  may  be  immediately  computed,  and  thus 
the  actual  amount  of  the  force  of  gravity  at  any  given  place 
may  be  ascertained. 

(220.)  To  compare  the  force  of  gravity  in  different  parts 


CHAP.    XI.  VARIATION    OP    GRAVITY.  133 

of  the  earth,  it  is  only  necessary  to  swing  the  same  pendu- 
lum in  the  places  under  consideration,  and  to  observe  the 
rapidity  of  its  vibrations.  The  proportion  of  the  force  of 
gravity  in  the  several  places  will  be  that  of  the  squares  of  the 
velocity  of  the  vibration.  Observations  to  this  effect  have 
been  made  at  several  places,  by  Biot,  Kater,  Sabine  and 
others. 

The  earth  being  a  mass  of  matter  of  a  form  nearly  spher- 
ical, revolving  with  considerable  velocity  on  an  axis,  its 
component  parts  are  affected  by  a  centrifugal  force  ;  in  virtue 
of  which,  they  have  a  tendency  to  fly  off  in  a  direction  per- 
pendicular to  the  axis.  This  tendency  increases  in  the  same 
proportion  as  the  distance  of  any  part  from  the  axis  increases, 
and  consequently  those  parts  of  the  earth  which  are  near  the 
equator,  are  more  strongly  affected  by  this  influence  than 
those  near  the  pole.  It  has  been  already  explained  (145)  that 
the  figure  of  the  earth  is  affected  by  this  cause,  and  that  it 
has  acquired  a  spheroidal  form.  The  centrifugal  force,  acting 
in  opposition  to  the  earth's  attraction,  diminishes  its  effects  ; 
and,  consequently,  where  this  fprce  is  more  efficient,  a  pendu- 
lum will  vibrate  more  slowly.  By  these  means  the  rate  of 
vibration  of  a  pendulum  becomes  an  indication  of  the  amount 
of  the  centrifugal  force.  But  this  latter  varies  in  proportion 
to  the  distance  of  the  place  from  the  earth's  axis ;  and  thus 
the  rate  of  a  pendulum  indicates  the  relation  of  the  distances 
of  different  parts  of  the  earth's  surface  from  its  axis.  The 
figure  of  the  earth  may  be  thus  ascertained,  and  that  which 
theory  assigns  to  it,  it  may  be  practically  proved  to  have. 

This,  however,  is  not  the  only  method  by  which  the  figure 
of  the  earth  may  be  determined.  The  meridians  being  sec- 
tions of  the  earth  through  its  axis,  if  their  figure  were  exactly 
determined,  that  of  the  earth  would  be  known.  Measure- 
ments of  arcs  of  meridians,  on  a  large  scale  have  been  exe- 
cuted, and  are  still  being  made  in  various  parts  of  the  earth, 
with  a  view  to  determine  the  curvature  of  a  meridian  at  dif- 
ferent latitudes.  This  method  is  independent  of  every  hy- 
pothesis concerning  the  density  and  internal  structure  of  the 
earth,  and  is  considered  by  some  to  be  susceptible  of  more 
accuracy  than  that  which  depends  on  the  observations  of 
pendulums. 

(221.)  It  has  been  stated  that,  when  the  arc  of  vibration 
of  a  pendulum  is  not  very  small,  a  variation  in  its  length  will 
produce  a  sensible  effect  on  the  time  of  vibration.  To  con- 
12 


THE    ELEMENTS    OF    MECHANICS.  CHAP.  XI. 

struct  a  pendulum  such  that  the  time  of  vibration  may  be 
independent  of  the  extent  of  the  swing,  was  a  favorite  spec- 
ulation of  geometers.  This  problem  was  solved  by  Huygens, 
who  showed  that  the  curve  called  a  cycloid,  previously  dis- 
covered and  described  by  Galileo,  possessed  the  isochronal 
property ;  that  is,  that  a  body  moving  in  it  by  the  force  of 
gravity  would  vibrate  in  the  same  time,  whatever  be  the 
length  of  the  arc  described. 

Let  O  A,f,g.  77.,  be  a  horizontal  line,  and  let  O  B  be  a 
circle  placed  below  this  line,  and  in  contact  with  it.  If  this 
circle  be  rolled  upon  the  line  from  O  towards  A,  a  point 
upon  its  circumference,  which,  at  the  beginning  of  the 
motion,  is  placed  at  O,  will,  during  the  motion,  trace  the 
curve  OCA.  This  curve  is  called  a  cycloid.  If  the  circle 
be  supposed  to  roll  in  the  opposite  direction  towards  A',  the 
same  point  will  trace  another  cycloid  O  C'  A'.  The  points 
C  and  C'  being  the  lowest  points  of  the  curves,  if  the  per- 
pendiculars C  D  and  C'  D'  be  drawn,  they  will  respectively 
be  equal  to  the  diameter  of  the  circle.  By  a  known  property 
of  this  curve,  the  arcs  O  C  and  O  C'  are  equal  to  twice  the 
diameter  of  the  circle.  From  the  point  O  suppose  a  flexible 
thread  to  be  suspended,  whose  length  is  twice  the  diameter 
of  the  circle, and  which  sustains  a  pendulous  body  P  at  its 
extremity.  If  the  curves  O  G  and  O  C',  from  the  plane  of 
the  paper,  be  raised  so  as  to  form  surfaces  to  which  the  thread 
may  be  applied,  the  extremity  P  will  extend  to  the  points  C 
and  C',  when  the  entire  thread  has  been  applied  to  either  of 
the  curves.  As  the  thread  is  deflected  on  either  side  of  its 
vertical  position,  it  is  applied  to  a  greater  or  lesser  portion  of 
either  curve,  according  to  the  quantity  of  its  deflection 
from  the  vertical.  If  it  be  deflected  on  each  side  until 
the  point  P  reaches  the  points  C  and  C',  the  extremity  would 
trace  a  cycloid  C  P  C'  precisely  equal  and  similar  to  those 
already  mentioned.  Availing  himself  of  this  property  of  the 
curve,  Huygens  constructed  his  cycloid al  pendulum.  The 
time  of  vibration  was  subject  to  no  variation,  however  the 
arc  of  vibration  might  change,  provided  only  that  the  length 
of  the  string  O  P  continued  the  same.  If  small  arcs  of  the 
cycloid  be  taken  on  either  side  of  the  point  P,  they  will 
not  sensibly  differ  from  arcs  of  a  circle  described  with 
the  centre  O  and  the  radius  O  P ;  for,  in  slight  deflections 
from  the  vertical  position,  the  effect  of  the  curves  O  C  and 
O  C'  on  the  thread  O  P  is  altogether  inconsiderable.  It  is 


CHAP.    XII.  SIMPLE    MACHINES.  135 

for  this  reason  that,  when  the  arcs  of  vibration  of  a  circular 
pendulum  are  small,  they  partake  of  the  property  of  isochro- 
nism  peculiar  to  those  of  a  cycloid.  But  when  the  deflection 
of  P  from  the  vertical  is  great,  the  effect  of  the  curves  O  C 
and  O  C'  on  the  thread  produces  a  considerable  deviation  of 
the  point  P  from  the  arc  of  the  circle  whose  centre  is  O  and 
whose  radius  is  O  P,  and  consequently  the  property  of 
isochronism  will  no  longer  be  observed  in  the  circular 
pendulum 


CHAPTER  XII. 

OF   SIMPLE    MACHINES. 

(222.)  A  MACHINE  is  an  instrument  by  which  force  or  mo- 
tion may  be  transmitted  and  modified  as  to  its  quantity  and 
direction.  There  are  two  ways  in  which  a  machine  may  be 
applied,  and  which  give  rise  to  a  division  of  mechanical  sci- 
ence into  parts  denominated  STATICS  and  DYNAMICS  ;  the 
one  including  the  theory  of  equilibrium,  and  the  other  the 
theory  of  motion.  When  a  machine  is  considered  statically, 
it  is  viewed  as  an  instrument  by  which  forces  of  determinate 
quantities  and  directions  are  made  to  balance  other  forces 
of  other  quantities  and  other  directions.  If  it  be  viewed 
dynamically,  it  is  considered  as  a  means  by  which  certain 
motions  of  determinate  quantity  and  direction  may  be  made 
to  produce  other  motions  in  other  directions  and  quantities. 
It  will  not  be  convenient,  however,  in  the  present  treatise, 
to  follow  this  division  of  the  subject.  We  shall,  on  the  other 
hand,  as  hitherto,  consider  the  phenomena  of  equilibrium 
and  motion  together. 

The  effects  of  machinery  are  too  frequently  described  in 
such  a  manner  as  to  invest  them  with  the  appearance  of  par- 
adox, and  to  excite  astonishment  at  what  appears  to  contra- 
dict the  results  of  the  most  common  experience.  It  will  be 
our  object  here  to  take  a  different  course,  and  to  attempt  to 
show  that  those  effects  which  have  been  held  up  as  matters 
of  astonishment  are  the  necessary,  natural  and  obvious  re- 
sults of  causes  adapted  to  produce  them  in  a  manner  analo- 
gous to  the  objects  of  most  familiar  experience. 

(223.)  In  the  application  of  a  machine  the^e  are  three 


136  THE    ELEMENTS    OF   MECHANICS.  CHAP.    XII. 

things  to  be  considered.  t>  The  force  or  resistance  which 
is  required  to  be  sustained,  opposed,  or  overcome.  2.  The 
force  which  is  used  to  sustain,  support,  or  overcome  that  re- 
sistance. 3.  The  machine  itself,  by  which  the  effect  of  this 
latter  force  is  transmitted  to  the  former.  Of  whatever  nature 
be  the  force  or  the  resistance  which  is  to  be  sustained  or 
overcome,  it  is  technically  called  the  weight^  since,  whatever 
it  be,  a  weight  of  equivalent  effect  may  always  be  found. 
The  force  which  is  employed  to  sustain  or  overcome  it  is 
technically  called  the  power. 

(224.)  In  expressing  the  effect  of  machinery,  it  is  usual  to 
say  that  the  power  sustains  the  weight ;  but  this,  in  fact,  is 
not  the  case,  and  hence  arises  that  appearance  of  paradox 
which  has  already  been  alluded  to.  If,  for  example,  it  is  said 
that  a  power  of  one  ounce  sustains  the  weight  of  one  ton, 
astonishment  is  not  unnaturally  excited,  because  the  fact,  as 
thus  stated,  if  the  terms  be  literally  interpreted,  is  physically 
impossible.  No  power  less  than  a  ton  can,  in  the  ordinary 
acceptation  of  the  word,  support  the  weight  of  a  ton.  It  will, 
however,  be  asked  how  it  happens  that  a  machine  appears  to 
do  this?  how  it  happens  that  by  holding  a  silken  thread, 
which  an  ounce  weight  would  snap,  many  hundred  weight 
may  be  sustained  ?  To  explain  this,  it  will  only  be  necessary 
to  consider  the  effect  of  a  machine,  when  the  power  and 
Weight  are  in  equilibrium. 

(225.)  In  every  machine  there  are  some  fixed  points  or 
props;  and  the  arrangement  of  the  parts  is  always  such,  that 
the  pressure,  excited  by  the  power  or  weight,  or  both,  is  dis- 
tributed among  these  props.  If  the  weight  amoumt  to  twenty 
hundred,  it  is  possible  so  to  distribute  it,  that  any  proportion, 
however  great,  of  it  may  be  thrown  on  the  fixed  points  or 
props  of  the  machine ;  the  remaining  part  only  can  properly 
be  said  to  be  supported  by  the  power  ;  and  this  part  can  never 
be  greater  than  the  power.  Considering  the  effect  in  this 
way,  it  appears  that  the  power  supports  just  so  much  of  the 
weight,  and  no  more,  as  is  equal  to  its  own  force,  and  that  all 
the  remaining  part  of  the  weight  is  sustained  by  the  machine. 

The  force  of  these  observations  will  be  more  apparent 
when  the  nature  and  properties  of  the  mechanic  powers  and 
other  machines  have  been  explained. 

(226.)  When  a  machine  is  used  dynamically /its  effects  are 
explained  on  different  principles.  It  is  true  that,  in  this  case, 
a  very  small  power  may  elevate  a  very  great  weight;  but, 


CHA^.  XII.  SIMPLE    MACHINES.  137 

nevertheless,  in  so  doing,  whatever  be  the  machine  used,  the 
total  expenditure  of  power,  in  raising  the  weight  through  any 
height,  is  never  less  than  that  which  would  be  expended  if 
the  power  were  immediately  applied  to  the  weight  without  the 
intervention  of  any  machine.  This  circumstance  arises  from 
an  universal  property  of  machines,  by  which  the  velocity  of 
the  weight  is  always  less  than  that  of  the  power,  in  exactly 
the  same  proportion  as  the  power  itself  is  less  than  the  weight ; 
so  that,  when  a  certain  power  is  applied  to  elevate  a  weight, 
the  rate  at  which  the  elevation  is  effecte^l  is  always  slow  in 
the  same  proportion  as  the  weight  is  great.  From  a  due  con- 
sideration of  this  remarkable  law,  it  will  easily  be  understood 
that  a  machine  can  never  diminish  the  total  expenditure  of 
power  necessary  to  raise  any  weight  or  to  overcome  any  re- 
sistance. In  such  cases,  all  that  a  machine  ever  does,  or  ever 
can  do,  is  to  enable  the  power  to  be  expended  at  a  slow  rate, 
and  in  a  more  advantageous  direction  than  if  it  were  immedi- 
ately applied  to  the  weight  or  the  resistance. 

Let  us  suppose  that  P  is  a  power  amounting  to  an  ounce, 
and  that  W  is  a  weight  amounting  to  50  ounces,  and  that  P 
elevates  W  by  means  of  a  machine.  In  virtue  of  the  prop- 
erty already  stated,  it  follows,  that  while  P  moves  through  50 
feet,  W  will  be  moved  through  1  foot ;  but  in  moving  P  through 
50  feet,  50  distinct  efforts  are  made,  by  each  of  which  1  ounce 
is  moved  through  1  foot,  and  by  which  collectively  50  distinct 
ounces  might  be  successively  raised  through  1  foot  But 
the  weight  W  is  50  ounces,  and  has  been  raised  through  1 
foot ;  from  whence  it  appears,  that  the  expenditure  of  power 
is  equal  to  that  which  would  be  necessary  to  raise  the  weight 
without  the  intervention  of  any  machine. 

This  important  principle  may  _be  presented  under  another 
aspect,  which  will  perhaps  render  it  more  apparent.  Suppose 
t  pi  weight  W  were  actually  divided  into  50  equal  parts,  or 
.suppose  it  were  a  vessel  of  liquid  weighing  50  ounces,  and 
containing  50  equal  measures;  if  these  50  measures'  were 
successively  lifted  through  a  height  of  1  foot,  the  efforts  neces- 
sary to  accomplish  this  would  be  the  same  as  tjhose  used  to 
move  the  power  P  through  50  feet,  and  it  is  obv  ious,  that  the 
total  expenditure  of  force  would  be  the  same  as  that  which 
would  be  necessary  to  lift  the  entire  contents  of  the  vessel 
through  1  foot. 

When  the  nature  and  properties  of  the  mechanic  powers 

and  other  machines  have  been  explained,  the  ji>rce  t?f  these 

12* 


138  THE    ELEMENTS    OF    MECHANICS.  CHAP.  XII. 

observations  will  be  more  distinctly  perceived,  The  effects 
of  props  and  fixed  points  in  sustaining  a  part  of  the  weight, 
and  sometimes  the  whole,  both  of  the  weight  and  power,  will 
then  be  manifest,  and  every  machine  will  furnish  a  verifica- 
tion of  the  remarkable  proportion  between  the  velocities  of 
the  weight  and  power,  which  has  enabled  us  to  explain  what 
might  otherwise  be  paradoxical  and  difficult  of  comprehen- 
sion. 

(227.)  The  most  simple  species  of  machines  are  those 
which  are  commonly  denominated  the  MECHANIC  POWERS. 
These  have  been  differently  enumerated  by  different  writers. 
If,  however,  the  object  be  to  arrange  in  distinct  classes,  and 
in  the  smallest  possible  number  of  them,  those  machines 
which  are  alike  in  principle,  the  mechanic  powers  may  be 
reduced  to  three. 

1.  The  lever. 

2.  The  cord. 

3.  The  inclined  plane 

To  one  or  other  of  these  classes  all  simple  machines  what- 
ever may  be  reduced,  and  all  complex  machines  may  be  re- 
solvefl  into  simple  elements  which  come  under  them. 

(228.)  The  first  class  includes  every  machine  which  is 
composed  of  a  solid  body  revolving  on  a  fixed  axis,  although 
the  name  lever  has  been  commonly  confined  to  cases  where 
the  machine  affects  certain  particular  forms.  This  is  by  far 
the  most  useful  class  of  machines,  and  will  require,  in  subse- 
quent chapters,  very  detailed  developement.  The  general 
principle,  uj>on  which  equilibrium  is  established  between 
the  power  and  weight  in  machines  of  this  class  has  been 
already  explained  in  (183.)  The  power  and  weight  are 
always  supposed  to  be  applied  in  directions  at  right  angles  to 
the  axis.  If  lines  be  drawn  from  the  axis  perpendicular  I* 
the  direction*  of  power  and  weight,  equilibrium  will  subsist, 
provided  the  power  multiplied  by  the  perpendicular  distance 
of  its  direction  from  the  axis,  be  equal  to  the  weight  multi- 
plied by  the  perpendicular  distance  of  its  direction  from  the 
axis.  This  is  a  principle  to  which  we  shall  have  occasion  to 
refer  in  explaining  the  various  machines  of  this  class. 

(229.)  If  the  moment  of  the  power  (KS4.)  be  greater  than 
that  of  the  weight,  the  effect  of  the  power  will  prevail  over 
that  of  the  weight,  and  elevate  it;  but  if,  on  the  other  hand, 
the  moment  of  the  uower  be  less  than  that  of  the  weight,  the 


CHAP,  XII.  SIMPLE    MACHINES.  139 

power  will  be  insufficient  to  support  the  weight,  and  will  allow 
it  to  fall. 

(230.)  The  second  class  of  simple  machines  includes  all 
those  cases  in  which  force  is  transmitted  by  means  of  flexible 
threads,  ropes,  or  chains.  The  principle,  by  which  the  effects 
of  these  machines  are  estimated,  is,  that  the  tension  through- 
out the  whole  length  of  the  same  cord,  provided  it  be  perfect- 
ly flexible,  and  free  from  the  effects  of  friction,  must  be  the 
same.  Thus,  if  a  force  acting  at  one  end  be  balanced  by  a 
force  acting  at  the  other  end,  however  the  cord  may  be  bent, 
or  whatever  course  it  may  be  compelled  to  take,  by  any  causes 
which  may  affect  it  between  its  ends,  these  forces  must  be 
equal,  provided  the  cord  be  free  to  move  over  any  obstacles 
which  may  deflect  it. 

Within  this  class  of  machines  are  included  all  the  various 
forms  of  pit/leys. 

(231.)  The  third  class  of  simple  machines  includes  all 
those  cases  in  which  the  weight  or  resistance  is  supported 
or  moved  on  a  hard  surface  inclined  to  the  vertical  direction. 

The  effects  of  such  machines  are  estimated  by  resolving 
the  whole  weight  of  the  body  into  two  elements  by  the  paral- 
lelogram of  forces.  One  of  these  elements  is  perpendicular 
to  the  surface,  and  supported  by  its  resistance ;  the  other  is 
parallel  to  the  surface,  and  supported  by  the  power.  The 
proportion,  therefore,  of  the  power  to  the  weight  will  always 
depend  on  the  obliquity  of  the  surface  to  the  direction  of  the 
weight.  This  will  be  easily  understood  by  referring  to  what 
has  been  already  explained  in  Chapter  VJlI. 

Under  this  class  of  machines,  come  the  inclined  plane, 
commonly  so  called,  the  wedge,  the  screw,  and  various 
others. 

(232.)  In  order  to  simplify  the  developement  of  the  ele- 
mentary theory  of  machines,  it  is  expedient  to  omit  the  con- 
sideration of  many  circumstances,  of  which,  however,  a  strict 
account  must  be  taken  before  any  practically  useful  applica- 
tion of  that  theory  can  be  attempted.  A  machine,  as  we 
must  for  the  present  contemplate  it,  is  a  thing  which  can  have 
no  real  or  practical  existence.  Its  various  parts  are  considered 
to  be  free  from  friction :  all  surfaces  which  move  in  contact 
are  supposed  to  be  infinitely  smooth  and  polished.  The  solid 
parts  are  conceived  to  be  absolutely  inflexible.  The  weight 
and  inertia  of  the  machine  itself  are  wholly  neglected,  arid 
we  reason  upon  it  as  if  it  were  divested  of  these  qualities. 


140  THE    ELEMENTS    OF    MECHANICS.  CHAP.  XII. 

Cords  and  ropes  are  supposed  to  have  no  stiffness,  to  be  infi- 
nitely flexible.  The  machine,  when  it  moves,  is  supposed  to 
suffer  no  resistance  from  the  atmosphere,  and  to  be  in  all  re- 
spects circumstanced  as  if  it  were'm  vacua. 

It  is  scarcely  necessary  to  state,  that,  all  these  suppositions 
being  false,  none  of  the  consequences  deduced  from  them  can 
be  true.  Nevertheless,  as  it  is  the  business  of  art  to  bring 
machines  as  near  to  this  state  of  ideal  perfection  as  possible, 
the  conclusions  which  are  thus  obtained,  though  false  in  a 
strict  sense,  yet  deviate  from  the  truth  in  but  a  small  degree. 
Like  the  first  outline  of  a  picture,  they  resemble,  in  their  gen- 
eral features,  that  truth  to  which,  after  many  subsequent  cor- 
rections, they  must  finally  approximate. 

After  a  first  approximation  has  been  made  on  the  several 
false  suppositions  which  have  been  mentioned,  various  effects, 
which  have  been  previously  neglected,  are  successively  taken 
into  account.  Roughness,  rigidity,  imperfect  flexibility,  the 
resistance  of  air  and  other  fluids,  the  effects  of  the  weight 
and  inertia  of  the  machine,  are  severally  examined,  and  their 
laws  and  properties  detected.  The  modifications  and  correc- 
tions, thus  suggested  as  necessary  to  be  introduced  into  our 
former  conclusions,  are  applied,  and  a  second  approximation, 
but  still  only  an  approximation,  to  truth  is  made.  For,  in  in- 
vestigating the  laws  which  regulate  the  several  effects  just 
mentioned,  we  are  compelled  to  proceed  upon  a  new  group 
of  false  suppositions.  To  determine  the  laws  which  regulate 
the  friction  of  surfaces,  it  is  necessary  to  assume  that  every 
part  of  the  surfaces  of  contact  is  uniformly  rough ;  that  the 
solid  parts  which  are  imperfectly  rigid,  and  the  cords  which 
are  imperfectly  flexible,  are  constituted  throughout  their  entire 
dimensions  of  a  uniform  material ;  so  that  the  imperfection 
does  not  prevail  more  in  one  part  than  another.  Thus  all  ir- 
regularity is  left  out  of  account,  and  a  general  average  of  the 
effects  taken.  It  is  obvious,  therefore,  that  by  these  means 
we  have  still  failed  in  obtaining  a  result  exactly  conformable 
to  the  real  state  of  things ;  but  it  is  equally  obvious  that  we 
have  obtained  one  much  more  conformable  to  that  state  than 
had  been  previously  accomplished,  and  sufficiently  near  it  for 
most  practical  purposes. 

This  apparent  imperfection  in  our  instruments  and  powers 
of  investigation  is  not  peculiar  to  mechanics :  it  pervades  all 
departments  of  natural  science.  In  astronomy,  the  motions 
of  the  celestial  bodies,  and  their  various  changes  and  appear 


CHAP.  XIII.  THE    LEVER.  141 

ances,  as  developed  by  theory,  assisted  by  observation  and 
experience,  are  only  approximations  to  the  real  motions  and 
appearances  which  take  place  in  nature.  It  is  true  that  these 
approximations  are  susceptible  of  almost  unlimited  accuracy ; 
but  still  they  are,  and  ever  will  continue  to  be,  only  approxi- 
mations. Optics  and  all  other  branches  of  natural  science 
are  liable  to  the  same  observations. 


CHAPTER  XIII. 

OF    THE    LEVER. 

(233.)  AN  inflexible,  straight  bar,  turning  on  an  axis,  is 
commonly  called  a  lever.  The  arms  of  the  lever  are  those 
parts  of  the  bar  which  extend  on  each  side  of  the  axis. 

The  axis  is  called  thejalcrum  or  prop. 

(234.)  Levers  are  commonly  divided  into  three  kinds,  ac- 
cording to  the  relative  positions  of  the  power,  the  weight  and 
the  fulcrum. 

In  a  lever  of  the  first  kind,  as  in  Jig.  78.,  the  fulcrum  is  be- 
tween the  power  and  weight. 

In  a  lever  of  the  second  kind,  as  in  Jig.  79.,  the  weight  is 
between  the  fulcrum  and  power. 

In  a  lever  of  the  third  kind,  as  in  Jig.  80.,  the  power  is  be- 
tween the  fulcrum  and  weight. 

(235.)  In  all  these  cases,  the  power  will  sustain  the  weight 
in  equilibrium,  provided  its  moment  be  equal  to  that  of  the 
weight.  (184.)  But  the  moment  of  the  power  is,  in  this  case, 
equal  to  the  product  obtained  by  multiplying  the  power  by  its 
distance  from  the  fulcrum,  and  the  moment  of  the  weight, 
by  multiplying  the  weight  by  its  distance  from  the  fulcrum. 
Thus,  if  the  number  of  ounces  in  P,  being  multiplied  by  the 
number  of  inches  in  P  F,  be  equal  to  the  number  of  ounces 
in  W,  multiplied  by  the  number  of  inches  in  W  F,  equilibrium 
will  be  established.  It  is  evident  from  this,  that  as  the  dis- 
tance of  the  power  from  the  fulcrum  increases  in  comparison 
to  the  distance  of  the  weight  from  the  fulcrum,  in  the  same 
degree  exactly  will  the  proportion  of  the  power  to  the  weight 
diminish.  In  other  words,  the  proportion  of  the  power  to  the 
weight  will  be  always  the  same  as  that  of  their  distances  from 
the  fulcrum  taken  in  a  reverse  order. 


142  THE    ELEMENTS  OF  MECHANICS.  CHAP.  XIII 

In  cases  where  a  small  power  is  required  to  sustain  or  ele- 
vate a  great  weight,  it  will  therefore  be  necessary  either  to 
remove  the  power  to  a  great  distance  from  the  fulcrum,  or  to 
bring  the  weight  very  near  it. 

(236.)  Numerous  examples  of  levers  of  the  first  kind  may 
be  given.  A  crow-bar,  applied  to  elevate  a  stone  or  other 
weight,  is  an  instance.  The  fulcrum  is  another  stone  placed 
near  that  which  is  to  be  raised,  and  the  power  is  the  hand 
placed  at  the  other  end  of  the  bar. 

A  handspike  is  a  similar  example. 

A  poker  applied  to  raise  fuel  is  a  lever  of  the  first  kind, 
the  fulcrum  being  the  bar  of  the  grate. 

Scissors,  shears,  nippers,  pincers,  and  other  similar  instru- 
ments, are  composed  of  two  levers  of  the  first  kind ;  the  ful- 
crum being  the  joint  or  pivot,  and  the  weight  the  resistance 
of  the  substance  to  be  cut  or  seized ;  the  power  being  the 
fingers  applied  at  the  other  end  of  the  levers. 

The  brake  of  a  pump  is  a  lever  of  the  first  kind  ;  the  pump- 
rods  and  piston  being  the  weight  to  be  raised. 

(237.)  Examples  of  levers  of  the  second  kind,  though  not 
so  frequent  as  those  just  mentioned,  are  not  uncommon. 

An  oar  is  a  lever  of  the  second  kind.  The  reaction  of 
the  water  against  the  blade  is  the  fulcrum.  The  boat  is  the 
weight,  and  the  hand  of  the  boatman  the  power. 

The  rudder  of  a  ship  or  boat  is  an  example  of  this  kind  of 
lever,  and  explained  in  a  similar  way. 

The  chipping  knife  is  a  lever  of  the  second  kind.  The 
end  attached  to  the  bench  is  the  fulcrum,  and  the  weight  the 
resistance  of  the  substance  to  be  cut,  placed  beneath  it. 

A  door  moved  upon  its  hinges  is  another  example. 

Nut-crackers  are  two  levers  of  the  second  kind ;  the  hinge 
which  unites  them  being  the  fulcrum,  the  resistance  of  the 
shell  placed  between  them  being  the  weight,  and  the  hand 
applied  to  the  extremity  being  the  power. 

A  wheelbarrow  is  a  lever  of  the  second  kind  ;  the  fulcrum 
being  the  point  at  which  the  wheel  presses  on  the  ground, 
arid  the  weight  being  that  of  the  barrow  and  its  load,  collect- 
ed at  their  centre  of  gravity. 

The  same  observation  may  be  applied  to  all  two-wheeled 
carriages,  which  are  partly  sustained  by  the  animal  which 
draws  them. 

(238.)  In  a  lever  of  the  third  kind,  the  weight,  being 
more  distant  from  the  fulcrum  than  the  power,  must  be  pro- 


CHAP.  XIII.  LEVERS.  143 

portionably  less  than  it.  In  this  instrument,  therefore,  the 
power  acts  upon  the  weight  to  a  mechanical  disadvantage, 
inasmuch  as  a  greater  power  is  necessary  to  support  or  move 
the  weight  than  would  be  required  if  the  power  were  imme- 
diately applied  to  the  weight,  without  the  intervention  of  a 
machine.  We  shall,  however,  hereafter  show  that  the  advan- 
tage which  is  lost  in  force  is  gained  in  despatch,  and  that 
in  proportion  as  the  weight  is  less  than  the  power  which  moves 
it,  so  will  the  speed  of  its  motion  be  greater  than  that  of  the 
power. 

Hence  a  lever  of  the  third  kind  is  only  used  in  cases 
where  the  exertion  of  great  power  is  a  consideration  subordi- 
nate to  those  of  rapidity  and  despatch. 

The  most  striking  example  of  levers  of  the  third  kind  is 
found  in  the  animal  economy.  The  limbs  of  animals  are 
generally  levers  of  this  description.  The  socket  of  the  bone 
is  the  fulcrum  ;  a  strong  muscle  attached  to  the  bone  near 
the  socket  is  the  power ;  and  the  weight  of  the  limb,  together 
with  whatever  resistance  is  opposed  to  its  motion,  is  the 
weight.  A  slight  contraction  of  the  muscle  in  this  case 
gives  a  considerable  motion  to  the  limb  :  this  effect  is  par- 
ticularly conspicuous  in  the  motion  of  the  arms  and  legs 
in  the  human  body  ;  a  very  inconsiderable  contraction  of 
the  muscles  at  the  shoulders  and  hips  giving  the  sweep  to 
the  limbs  from  which  the  body  derives  so  much  activity. 

The  treddle  of  the  turning  lathe  is  a  lever  of  the  third 
kind.  The  hinge  which  attaches  it  to  the  floor  is  the  ful- 
crum, the  foot  applied  to  it  near  the  hinge  is  the  power,  and 
the  crank  upon  the  axis  of  the  fly-wheel,  with  which  its  ex- 
tremity is  connected,  is  the  weight. 

Tongs  are  levers  of  this  kind,  as  also  the  shears  used  in 
shearing  sheep.  In  these  cases,  the  power  is  the  hand  placed 
immediately  below  the  fulcrum,  or  point  where  the  two  levers 
are  connected. 

(239.)  When  the  power  is  said  to  support  the  weight  by 
means  of  a  lever,  or  any  other  machine,  it  is  only  meant  that 
the  power  keeps  the  machine  in  equilibrium,  and  thereby 
enables  it  to  sustain  the  weight.  It  is  necessary  to  attend 
to  this  distinction,  to  remove  the  difficulty  which  may  arise 
from  the  paradox  of  a  small  power  sustaining  a  great  weight. 

In  a  lever  of  the  first  kind,  the  fulcrum  F,  Jig.  78.,  or 
axis,  sustains  the  united  forces  of  the  power  and  weight. 

In  a  lever  of  the  second  kind,  if  the  power  be  supposed  to 


144  THE    ELEMENTS    OF    MECHANICS.  CHAP.  XIII. 

act  over  a  wheel  R,  fig.  79.,  the  fulcrum  F  sustains  a  pres- 
sure equal  to  the  difference  between  the  power  and  weight, 
and  the  axis  of  the  wheel  R  sustains  a  pressure  equal  to 
twice  the  power  ;  so  that  the  total  pressures  on  F  and  R  are 
equivalent  to  the  united  forces  of  the  power  and  weight. 

In  a  lever  of  the  third  kind  similar  observations  are  appli- 
cable. The  wheel  R,  jig.  80.,  sustains  a  pressure  equal  to 
twice  the  power,  and  the  fulcrum  F  sustains  a  pressure  equal 
to  the  difference  between  the  power  and  weight. 

These  facts  may  be  experimentally  established  by  attach- 
ing a  string  to  the  lever  immediately  over  the  fulcrum,  and 
suspending  the  lever  by  that  string  from  the  arm  of  a  balance. 
The  counterpoising  weight,  when  the  fulcrum  is  removed, 
will,  in  the  first  case,  be  equal  to  the  sum  of  the  weight 
and  power,  and  in  the  last  two  cases  equal  to  their  differ- 
ence. 

(240.)  We  have  hitherto  omitted  the  consideration  of  the 
effect  of  the  weight  of  the  lever  itself.  If  the  centre  of 
gravity  of  the  lever  be  in  the  vertical  line  through  the  axis, 
the  weight  of  the  instrument  will  have  no  other  effect  than 
to  increase  the  pressure  on  the  axis  by  its  own  amount.  But 
if  the  centre  of  gravity  be  on  the  same  side  of  the  axis  with 
the  weight,  as  at  G,  it  will  oppose  the  effect  of  the  power, 
a  certain  part  of  which  must  therefore  be  allowed  to  support 
it.  To  ascertain  what  part  of  the  power  is  thus  expended, 
it  is  to  be  considered  that  the  moment  of  the  weight  of  the 
lever  collected  at  G,  is  found  by  multiplying  that  weight  by 
the  distance  G  F.  The  moment  of  that  part  of  the  power 
which  supports  this  must  be  equal  to  it ;  therefore,  it  is  only 
necessary  to  find  how  much  of  the  power  multiplied  by  P  F 
will  be  equal  to  the  weight  of  the  lever  multiplied  by  G  F. 
This  is  a  question  in  common  arithmetic. 

If  the  centre  of  gravity  of  the  lever  be  at  a  different  side 
of  the  axis  from  the  weight,  as  at  G',the  weight  of  the  instru- 
ment will  co-operate  with  the  power  in  sustaining  the  weight 
W.  To  determine  what  portion  of  the  weight  W  is  thus 
sustained  by  the  weight  of  the  lever,  it  is  only  necessary  to 
find  how  much  of  W,  multiplied  by  the  distance  W  F,  is 
equal  to  the  weight  of  the  lever  multiplied  by  G'  F. 

In  these  cases,  the  pressure  on  the  fulcrum,  as  already 
estimated,  will  always  be  increased  by  the  weight  of  the 
Jever. 

(241.)  The  sense  in  which  a  small  power  is  said  to  sustain 


CHAP.  XIII.  LEVERS.  145 

a  great  weight,  and  the  manner  of  accomplishing  this,  being 
explained,  we  shall  now  consider  how  the  power  is  applied 
in  moving  the  weight.  Let  P  W,  Jig.  81.,  be  the  places 
of  the  power  and  weight,  and  F  that  of  the  fulcrum,  and  let 
the  power  be  depressed  to  P'  while  the  weight  is  raised  to 
W.  The  space  P  P'  evidently  bears  the  same  proportion  to 
W  W',  as  the  arm  P  F  to  W  F.  Thus,  if  P  F  be  ten  times 
W  F,  P  P'  will  be  ten  times  W  W'.  A  power  of  one  pound 
at  P  being  moved  from  P  to  P7,  will  carry  a  weight  of  ten 
pounds  from  W  to  W.  But  in  this  case  it  ought  not  to  be 
said,  that  a  lesser  weight  moves  a  greater,  for  it  is  not  diffi- 
cult to  show  that  the  total  expenditure  of  force  in  the  motion 
of  one  pound  from  P  to  P7  is  exactly  the  same  as  in  the  mo- 
tion of  ten  pounds  from  W  to  W7-  If  the  space  P  P7  be  ten 
inches,  the  space  W  W7  will  be  one  inch.  A  weight  of  one 
pound  is  therefore  moved  through  ten  successive  inches,  and 
in  each  inch  the  force  expended  is  that  which  would  be  suffi- 
cient to  move  one  pound  through  one  inch.  The  total  expen- 
diture of  force  from  P  to  P7  is  ten  times  the  force  necessary 
to  move  one  pound  through  one  inch,  or,  what  is  the  same, 
it  is  that  which  would  be  necessary  to  move  ten  pounds  through 
one  inch.  But  this  is  exactly  what  is  accomplished  by  the 
opposite  end  W  of  the  lever  ;  for  the  weight  W  is  ten  pounds, 
and  the  space  W  W  is  one  inch. 

If  the  \veight  W  of  ten  pounds  could  be  conveniently  di- 
vided into  ten  equal  parts  of  one  pound  each,  each  part  might 
be  separately  raised  through  one  inch,  without  the  interven- 
tion of  the  lever  or  any  other  machine.  In  this  case,  the 
same  quantity  of  power  would  be  expended,  and  expended  in 
the  same  manner  as  in  the  case  just  mentioned. 

It  is  evident,  therefore,  that  when  a  machine  is  applied  to 
raise  a  weight  or  to  overcome  resistance,  as  much  force  must 
be  really  used  as  if  the  power  were  immediately  applied  to 
the  weight  or  resistance.  All  that  is  accomplished  by  the 
machine  is  to  enable  the  power  to  do  that  by  a  succession 
of  distinct  efforts  which  should  be  otherwise  performed  by  a 
single  effort.  These  observations  will  be  found  to  be  appli- 
cable to  all  other  machines. 

(242.)  Weighing  machines  of  almost  every  kind,  whether 
used  for  commercial  or  philosophical  purposes,  are  varieties 
of  the  lever.  The  common  balance,  which,  of  all  weighing 
machines,  is  the  most  perfect,  and  best  adapted  for  ordinary 
use  whether  in  commerce  or  experimental  philosophy,  is  a 
13 


146  THE  ELEMENTS  OF  MECHANICS.  CHAP.  XIII. 

lever  with  equal  arms.  In  the  steel-yard,  one  weight  serves 
as  a  counterpoise  and  measure  of  others  of  different  amount, 
by  receiving  a  leverage  variable  according  to  the  varying 
amount  of  the  weight  against  which  it  acts.  A  detailed 
account  of  such  instruments  will  be  found  in  Chapter  XXI. 

(243.)  We  have  hitherto  considered  the  power  and  weight 
as  acting  on  the  lever,  in  directions  perpendicular  to  its 
length,  and  parallel  to  each  other.  This  does  not  always 
happen.  Let  A  B,  Jig.  83.,  be  a  lever  whose  fulcrum  is  F, 
and  let  A  R  be  the  direction  of  the  power,  and  B  S  the 
direction  of  the  weight.  If  the  lines  R  A  and  S  B  be  con- 
tinued, and  perpendiculars  F  C  and  F  D  drawn  from  the 
fulcrum  to  those  lines,  the  moment  of  the  power  will  be  found 
by  multiplying  the  power  by  the  line  F  C,  and  the  moment 
of  the  weight  by  multiplying  the  weight  by  F  D.  If  these 
moments  be  equal,  the  power  will  sustain  the  weight  in  equi- 
librium. (185.) 

It  is  evident  that  the  same  reasoning  will  be  applicable 
when  the  arms  of  the  lever  are  not  in  the  same  direction. 
These  arms  may  be  of  any  figure  or  shape,  and  may  be  placed 
relatively  to  each  other  in  any  position. 

(244.)  In  the  rectangular  lever  the  arms  are  perpendicular 
to  each  other,  and  the  fulcrum  F,  Jig.  84.,  is  at  the  right 
angle.  The  moment  of  the  power,  in  this  case,  is  P  multi- 
plied by  A  F,  and  that  of  the  weight  W  multiplied  by  B  F. 
When  the  instrument  is  in  equilibrium  these  moments  must 
be  equal. 

When  the  hammer  is  used  for  drawing  a  nail,  it  is  a  lever 
of  this  kind  :  the  claw  of  the  hammer  is  the  shorter  arm  ; 
the  resistance  of  the  nail  is  the  weight ;  and  the  hand  ap- 
plied to  the  handle  the  power. 

(245.)  When  a  beam  rests  on  two  props  A  B,  Jig.  85., 
and  supports,  at  some  intermediate  place  C,  a  weight  W, 
this  weight  is  distributed  between  the  props  in  a  manner 
which  may  be  determined  by  the  principles  already  explained. 
If  the  pressure  on  the  prop  B  be  considered  as  a  power  sus- 
taining the  weight  W,  by  means  of  the  lever  of  the  second 
kind  B  A,  then  this  power  multiplied  by  B  A  must  be  equal 
to  the  weight  multiplied  by  C  A.  Hence  the  pressure  on  B 
will  be  the  same  fraction  of  the  weight  as  the  part  A  C  is 
of  A  B.  In  the  same  manner  it  may  be  proved,  that  the 
pressure  on  A  is  the  same  fraction  of  the  weight  as  B  C  is 
of  B  A.  Thus,  if  A  C  be  one  third,  and  therefore  B  C  two 


CHAP.  XIII.  COMPOUND  LEVERS.  147 

thirds  of  B  A,  the  pressure  on  B  will  be  one  third  of  the 
weight,  and  the  pressure  on  A  two  thirds  of  the  weight. 

It  follows  from  this  reasoning,  that  if  the  weight  be  in  the 
middle,  equally  distant  from  B  and  A,  each  prop  will  sustain 
half  the  weight.  The  effect  of  the  weight  of  the  beam  itself 
may  be  determined  by  considering  it  to  be  collected  at  its 
centre  of  gravity.  If  this  point,  therefore,  be  equally  distant 
from  the  props,  the  weight  of  the  beam  will  be  equally  dis- 
tributed between  them. 

According  to  these  principles,  the  manner  in  which  a  load 
borne  on  poles  between  two  bearers  is  distributed  between 
them  may  be  ascertained.  As  the  efforts  of  the  bearers  and 
th«  direction  of  the  weight  are  always  parallel,  the  position 
of  the  poles  relatively  to  the  horizon  makes  no  difference  in 
the  distribution  of  the  weights  between  the  bearers.  Whether 
they  ascend  or  descend,  or  move  on  a  level  plane,  the  weight 
will  be  similarly  shared  between  them. 

If  the  beam  extend  beyond  the  prop,  as  in  Jig.  86.,  and 
the  weight  be  suspended  at  a  point  not  placed  between 
them,  the  props  must  be  applied  at  different  sides  of  the  beam. 
The  pressures  which  they  sustain  may  be  calculated  in  the 
same  manner  as  in  the  former  case.  The  pressure  of  the 
prop  B  may  be  considered  as  a  power  sustaining  the  weight 
W  by  means  of  the  lever  B  C.  Hence  the  pressure  of  B, 
multiplied  by  B  A,  must  be  equal  to  the  weight  W  multiplied 
by  A  C.  Therefore  the  pressure  on  B  bears  the  same  pro- 
portion to  the  weight  as  A  C  does  to  A  B.  In  the  same 
manner,  considering  B  as  a  fulcrum,  and  the  pressure  of  the 
prop  A  as  the  power,  it  may  be  proved  that  the  pressure  of  A 
bears  the  same  proportion  to  the  weight  as  the  line  B  C  does 
to  A  B.  It  therefore  appears,  that  the  pressure  on  the  prop 
A  is  greater  than  the  weight. 

(•246.)  When  great  power  is  required,  and  it  is  incon- 
venient to  construct  a  long  lever,  a  combination  of  levers 
may  be  used.  In  Jig.  87.  such  a  system  of  levers  is  repre- 
sented, consisting  of  three  levers  of  the  first  kind.  The  man- 
ner in  which  the  effect  of  the  power  is  transmitted  to  the 
weight  may  be  investigated  by  considering  the  effect  of  each 
lever  successively.  The  power  at  P  produces  an  upward  force 
at  P',  which  bears  to  P  the  same  proportion  as  P'  F  to  P  F. 
Therefore  the  effect  at  P'  is  as  many  times  the  power  as  the 
line  P  F  is  of  P'  F.  Thus,  if  P  F  be  ten  times  P'  F,  the 
upward  force  at  P'  is  ten  times  the  power.  The  arm  P'  F' 


148  THE  ELEMENTS  OF  MECHANICS.  CHAP.  XIII. 

of  the  second  lever  is  pressed  upwards  by  a  force  equal  to 
ten  times  the  power  at  P.  In  the  same  manner  this  may  be 
shown  to  produce  an  effect  at  P"  as  many  times  greater 
than  P'  as  P>  F'  is  greater  than  P"  F'.  Thus,  if  P'  F'  be 
twelve  times  P"  F7,  the  effect  at  P"  will  be  twelve  times 
that  of  P'.  But  this  last  was  ten  times  the  power,  and  there- 
fore the  P"  will  be  one  hundred  and  twenty  times  the  power. 
In  the  same  manner  it  may  be  shown  that  the  weight  is  as 
many  times  greater  than  the  effect  at  P"  r-s  P"  F"  is  greater 
than  W  F".  If  P"  F"  be  five  times  W  F",  the  weight  will 
be  five  times  the  effect  at  P".  But  this  effect  is  one  hundred 
and  twenty  times  the  power,  and  therefore  the  weight  would 
be  six  hundred  times  the  power. 

In  the  same  manner  the  effect  of  any  compound  system 
of  levers  may  be  ascertained  by  taking  the  proportion  of  the 
weight  to  the  power  in  each  lever  separately,  and  multiplying 
these  numbers  together.  In  the  example  given,  these  pro- 
portions are  10,  12,  and  5,  which  multiplied  together  give 
600.  In  jig.  87.  the  levers  composing  the  system  are  of  the 
first  kind  ;  but  the  principles  of  the  calculation  will  not  be 
altered  if  they  be  of  the  second  or  third  kind,  or  some  of 
one  kind  and  some  of  another. 

(247.)  That  number  which  expresses  the  proportion  of  the 
weight  to  the  equilibrating  power  in  any  machine,  we  shall 
call  the  power  of  the  machine.  Thus,  if,  in  a  lever,  a  power 
of  one  pound  support  a  weight  of  ten  pounds,  the  power  of 
the  machine  is  ten.  If  a  power  of  2  Ibs.  support  a  weight  of 
11  Ibs.,  the  power  of  the  machine  is  5£,  2  being  contained  in 
11  5^  times. 

(248.)  As  the  distances  of  the  power  and  weight  from  the 
fulcrum  of  a  lever  may  be  varied  at  pleasure,  and  any  assign- 
ed proportion  given  to  them,  a  lever  may  always  be  conceived 
having  a  power  equal  to  that  of  any  given  machine.  Such 
a  lever  may  be  called,  in  relation  to  that  machine,  the  equiv- 
alent lever. 

As  every  complex  machine  consists  of  a  number  of  simple 
machines  acting  one  upon  another,  and  as  each  simple  ma- 
chine may  be  represented  by  an  equivalent  lever,  the  complex 
machine  will  be  represented  by  a  compound  system  of  equiv- 
alent levers.  From  what  has  been  proved  in  (246.),  it  there- 
fore follows  that  the  power  of  a  complex  machine  may  be 
calculated  by  multiplying  together  the  powers  of  the  several 
simple  machines  of  which  it  is  composed. 


CHAP.    XIV.  WHEEL-WORK.  149 

CHAPTER   XIV. 

OF     WHEEL-WORK. 

(249.)  WHEN  a  lever  is  applied  to  raise  a  weight,  or  over- 
come a  resistance,  the  space  through  which  it  acts  at  any  one 
time  is  small,  and  the  work  must  be  accomplished  by  a  suc- 
cession of  short  and  intermitting  efforts.  In  Jig.  81.,  after 
the  weight  has  been  raised  from  W  to  W,  the  lever  must 
again  return  to  its  first  position,  to  repeat  the  action.  During 
this  return  the  motion  of  the  weight  is  suspended,  and  it  will 
fall  downwards  unless  some  provision  be  made  to  sustain  it. 
The  common  lever  is,  therefore,  only  used  in  cases  where 
weights  are  required  to  be  raised  through  small  spaces,  and 
under  these  circumstances  its  great  simplicity  strongly  rec- 
ommends it.  But  where  a  continuous  motion  is  to  be  pro- 
duced, as  in  raising  ore  from  the  mine,  or  in  weighing  the 
anchor  of  a  vessel,  some  contrivance  must  be  adopted  to  re- 
move the  intermitting  action  of  the  lever,  and  render  it  con- 
tinual. The  various  forms  given  to  the  lever,  with  a  view  to 
accomplish  this,  are  generally  denominated  the  wheel  and  axle . 

In  jig.  88.,  A  B  is  a  horizontal  axle,  which  rests  in  pivots 
at  its  extremities,  or  is  supported  in  gudgeons,  and  capable  of 
revolving.  Round  this  axis  a  rope  is  coiled,  which  sustains 
the  weight  W.  On  the  same  axis  a  wheel  C  is  fixed,  round 
which  a  rope  is  coiled  in  a  contrary  direction,  to  which  is 
appended  the  power  P.  The  moment  of  the  power  is  found 
by  multiplying  it  by  the  radius  of  a  wheel,  and  the  moment  of 
the  weight  by  multiplying  it  by  the  radius  of  its  axle.  If  these 
moments  be  equal  (185),  the  machine  will  be  in  equilibrium. 
Whence  it  appears  that  the  power  of  the  machine  (247.)  is 
expressed  by  the  proportion  which  the  radius  of  the  wheel 
bears  to  the  radius  of  the  axle :  or,  what  is  the  same,  of  the 
diameter  of  the  wheel  to  the1*  diameter  of  the  axle.  This 
principle  is  applicable  to  the  wheel  and  axle  in  every  variety 
of  form  under  which  it  can  be  presented. 

(250.)  It  is  evident  that  as  the  power  descends  continually, 
and  the  rope  is  uncoiled  from  the  wheel,  the  weight  will  be 
raised  continually,  the  rope  by  which  it  is  suspended  being  at 
the  same  time  coiled  upon  the  axle. 

When  the  machine  is  in  equilibrium,  the  forces  of  both 
the  weight  and  power  are  sustained  by  the  axle,  and  dis- 
13  * 


150  THE    ELEMENTS    OP   MECHANICS.  CHAP.    XIV. 

tributed   between    its   props,   in   the   manner   explained    in 


When  the  machine  is  applied  to  raise  a  weight,  the  velocity 
with  which  the  power  moves  is  as  many  times  greater  than 
that  with  which  the  weight  rises,  as  the  weight  itself  is  great- 
er than  the  power.  This  is  a  principle  which  has  already 
been  noticed,  and  which  is  common  to  all  machines  whatso- 
ever. It  may  hence  be  proved,  that  in  the  elevation  of  the 
weight  a  quantity  of  power  is  expended  equal  to  that  which 
would  be  necessary  to  elevate  the  weight  if  the  power  were 
immediately  applied  to  it,  without  the  intervention  of  any 
machine.  This  has  been  explained  in  the  case  of  the  lever 
in  (241.),  and  may  be  explained  in  the  present  instance  in 
nearly  the  same  words. 

In  one  revolution  of  the  machine  the  length  of  rope  un- 
coiled from  the  wheel  is  equal  to  the  circumference  of  the 
wheel,  and  through  this  space  the  power  must  therefore  move. 
At  the  same  time  the  length  of  rope  coiled  upon  the  axle  is 
equal  to  the  circumference  of  the  axle,  and  through  this  space 
the  weight  must  be  raised.  The  spaces,  therefore,  through 
which  the  power  and  weight  move  in  the  same  time,  are  in 
the  proportion  of  the  circumferences  of  the  wheel  and  axle ; 
but  these  circumferences  are  in  the  same  proportion  as  their 
diameters.  Therefore  the  velocity  of  the  power  will  bear  to 
the  velocity  of  the  weight  the  same  proportion  as  the  diame- 
ter of  the  wheel  bears  to  the  diameter  of  the  axle,  or,  what  is 
the  same,  as  the  weight  bears  to  the  power.  (249.) 

(251.)  We  have  here  omitted  -the  consideration  of  the 
thickness  of  the  rope.  When  this  is  considered,  the  force 
must  be  conceived  as  acting  in  the  direction  of  the  centre  of 
the  rope,  and  therefore  the  thickness  of  the  rope  which  sup- 
ports the  power  ought  to  be  added  to  the  diameter  of  the 
wheel,  and  the  thickness  of  the  rope  which  supports  the 
weight  to  the  diameter  of  the  axle.  It  is  the  more  necessary  to 
attend  to  this  circumstance,  as"  the  strength  of  the  rope  neces- 
sary to  support  the  weight  causes  its  thickness  to  bear  a  con- 
siderable proportion  to  the  diameter  of  the  axle ;  while  the 
rope  which  sustains  the  power  not  requiring  the  same  strength, 
and  being  applied  to  a  larger  circle,  bears  a  very  inconsidera- 
ble proportion  to  its  diameter. 

(252.)  In  numerous  forms  of  the  wheel  and  axle,  the  weight 
or  resistance  is  applied  by  a  rope  coiled  upon  the  axle ;  but 
the  manner  in  which  the  power  is  applied  is  very  various,  and 


CHAP.  XIV.   WINDLASS CAPSTAN TREADMILL.         151 

not  often  by  means  of  a  rope.  The  circumference  of  a  wheel 
sometimes  carries  projecting  pins,  as  represented  in  Jig.  88., 
to  which  the  hand  is  applied  to  turn  the  machine.  An  in- 
stance of  this  occurs  in  the  wheel  used  in  the  steerage  of  a 
vessel. 

In  the  common  windlass,  the  power  is  applied  by  means  of 
a  winch,  which  is  a  rectangular  lever,  as  represented  in  Jig. 
89.  The  arm  B  C  of  the  winch  represents  the  radius  of  the 
Awheel,  and  the  power  is  applied  to  C  D  at  right  angles  to 
B  C. 

In  some  cases,  no  wheel  is  attached  to  the  axle ;  but  it  is 
pierced  with  holes  directed  towards  its  centre,  in  which  long 
levers  are  incessantly  inserted,  and  a  continuous  action  pro- 
duced by  several  men  working  at  the  same  time ;  so  that 
while  some  are  transferring  the  levers  from  hole  to  hole,  others 
are  working  the  windlass. 

The  axle  is  sometimes  placed  in  a  vertical  position,  the 
wheel  or  levers  being  moved  horizontally.  The  capstan  is  an 
example  of  this :  a  vertical  axis  is  fixed  in  the  deck  of  the 
ship ;  the  circumference  is  pierced  with  holes  presented  to- 
wards its  centre.  These  holes  receive  long  levers,  as  rep- 
resented in  jig.  90.  The  men  who  work  the  capstan  walk 
continually  round  the  axle,  pressing  forward  the  levers  near 
their  extremities. 

In  some  cases,  the  wheel  is  turned  by  the  weight  of  animals 
placed  at  its  circumference,  who  move  forward  as  fast  as  the 
wheel  descends,  so  as  to  maintain  their  position  continually 
at  the  extremity  of  the  horizontal  diameter.  The  treadmill 
fig.  91.,  and  certain  cranes,  such  &sjig.  92.,  are  examples  of 
this. 

In  water-wheels,  the  power  is  the  weight  of  water  contain- 
ed in  buckets  at  the  circumference,  as  in  Jig.  93.,  which  is 
called  an  over-shot  wheel ;  and  sometimes  the  impulse  of 
water  against  float-boards  at  the  circumference,  as  in  the 
under-shot  wheel,  Jig.  94.  Both  these  principles  act  in  the 
breast-wheel,  ^/zg-.  95. 

In  the  paddle-wheel  of  a  steamboat,  the  power  is  the  re- 
sistance which  the  water  offers  to  the  motion  of  the  paddle- 
boards. 

In  windmills,  the  power  is  the  force  of  the  wind  acting  on 
various  parts  of  the  arms,  and  may  be  considered  as  different 
powers  simultaneously  acting  on  different  wheels  having  the 
same  axle. 


152  THE    ELEMENTS    OF    MECHANICS.  CHAP.    XIV. 

(253.)  In  most  cases  in  which  the  wheel  and  axle  is  used, 
the  action  of  the  power  is  liable  to  occasional  suspension  or 
intermission,  in  which  case  some  contrivance  is  necessary  to 
prevent  the  recoil  of  the  weight.  A  ratchet  wheel  H,  Jig.  88., 
is  provided  for  this  purpose,  which  is  a  contrivance  which  per- 
mits the  wheel  to  turn  in  one  direction ;  but  a  catch  which 
falls  between  the  teeth  of  a  fixed  wheel,  prevents  its  motion 
in  the  other  direction.  The  effect  of  the  power  or  weight  is 
sometimes  transmitted  to  the  wheel  or  axle  by  means  of  a 
straight  bar,  on  the  edge  of  which  teeth  are  raised,  which 
engage  themselves  in  corresponding  teeth  on  the  wheel  or  axle. 
Such  a  bar  is  called  a  rack  :  and  an  instance  of  its  use  may 
be  observed  in  the  manner  of  working  the  pistons  of  an  air- 
pump. 

(254.)  The  power  of  the  wheel  and  axle  being  expressed 
by  the  number  of  times  the  diameter  of  the  axle  is  contained 
in  that  of  the  wheel,  there  are  obviously  only  two  ways  by 
which  this  power  may  be  increased  ;  viz.  either  by  diminishing 
the  diameter  of  the  axle,  or  increasing  that  of  the  wheel. 
In  cases  where  great  power  is  required,  each  of  these  methods 
is  attended  with  practical  inconvenience  and  difficulty.  If 
the  diameter  of  the  wheel  be  considerably  enlarged,  the  ma- 
chine will  become  unwieldy,  and  the  power  will  work  through 
an  unmanageable  space.  If,  on  the  other  hand,  the  power  of 
the  machine  be  increased  by  reducing  the  thickness  of  the 
axle,  the  strength  of  the  axle  will  become  insufficient  for  the 
support  of  that  weight,  the  magnitude  of  which  had  render- 
ed the  increase  of  the  power  of  the  machine  necessary.  To 
combine  the  requisite  strength  with  moderate  dimensions  and 
great  mechanical  power,  is,  therefore,  impracticable  in  the 
ordinary  form  of  the  wheel  and  axle.  This  has,  however, 
been  accomplished  by  giving  different  thicknesses  to  different 
parts  of  the  axle,'  and  carrying  a  rope,  which  is  coiled  on  the 
thinner  part,  through  a  wheel  attached  to  the  weight,  and  coil- 
ing it  in  the  opposite  direction  on  the  thicker  part,  as  \njig. 
96.  To  investigate  the  proportion  of  the  power  to  the  weight 
in  this  case,  let  Jig.  97.  represent  a  section  of  the  apparatus 
at  right  angles  to  the  axis.  The  weight  is  equally  suspended 
by  the  two  parts  of  the  rope,  S  and  S',  and  therefore  each 
part  is  stretched  by  a  force  equal  to  half  the  weight.  The 
moment  of  the  force,  which  stretches  the  rope  S,  is  half  the 
weight  multiplied  by  the  radius  of  the  thinner  part  of  the  axle. 
This  force,  being  at  the  same  side  of  the  centre  with  the  pow- 


CHAP.  XIV.          COMPOUND  AXLE.  153 

er,  co-operates  with  it  in  supporting  the  force  which  stretches 
S',  and  which  acts  at  the  other  side  of  the  centre.  By  the 
principle  established  in  (185.),  the  moments  of  P  and  S  must 
be  equal  to  that  of  S' :  and  therefore  if  P  be  multiplied  by 
the  radius  of  the  wheel,  and  added  to  half  the  weight  multi- 
plied by  the  radius  of  the  thinner  part  of  the  axle,  we  must 
obtain  a  sum  equal  to  half  the  weight  multiplied  by  the  radius 
of  the  thicker  part  of  the  axle.  Hence  it  is  easy  to  perceive, 
that  the  power  multiplied  by  the  radius  of  the  wheel  is  equal 
to  half  the  weight  multiplied  by  the  difference  of  the  radii 
of  the  thicker  and  thinner  parts  of  the  axle ;  or,  what  is  the 
same,  the  power  multiplied  by  the  diameter  of  the  wheel 
is  equal  to  the  weight  multiplied  by  half  the  difference  of  the 
diameters  of  the  thinner  and  thicker  parts  of  the  axle. 

A  wheel  and  axle  constructed  in  this  manner  is  equivalent 
to  an  ordinary  one,  in  which  the  wheel  has  the  same  diameter, 
and  whose  axle  has  a  diameter  equal  to  half  the  difference  of 
the  diameters  of  the  thicker  and  thinner  parts.  The  power 
of  the  machine  is  expressed  by  the  proportion  which  the  diam- 
eter of  the  wheel  bears  to  half  the  difference  of  these  diam- 
eters ;  and  therefore  this  power,  when  the  diameter  of  the 
wheel  is  given,  does  not,  as  in  the  ordinary  wheel  and  axle, 
depend  on  the  smallness  of  the  axle,  but  on  the  smallness  of 
the  difference  of  the  thinner  and  thicker  parts  of  it.  The 
axle  may,  therefore,  be  constructed  of  such  a  thickness  as  to 
give  it  all  the  requisite  strength,  and  yet  the  difference  of  the 
diameters  of  its  different  parts  may  be  so  small  as  to  give  it 
all  the  requisite  power. 

(255.)  It  often  happens  that  a  varying  weight  is  to  be  rais- 
ed, or  resistance  overcome,  by  a  uniform  power.  If,  in  such 
a  case,  the  weight  be  raised  by  a  rope  coiled  upon  a  uniform 
axle,  the  action  of  the  power  would  not  be  uniform,  but  would 
vary  with  the  weight.  It  is,  however,  in  most  cases  desirable 
or  necessary  that  the  weight  or  resistance,  even  though  it  vary, 
shall  be  moved  uniformly.  This  will  be  accomplished  if  by 
any  means  the  leverage  of  the  weight  is  made  to  increase  in 
the  same  proportion  as  the  weight  diminishes,  and  to  dimin- 
ish in  the  same  proportion  as  the  weight  increases ;  for  in  that 
case  the  moment  of  the  weight  will  never  vary,  whatever  it 
gains  by  the  increase  of  weight  being  lost  by  the  diminished 
leverage,  and  whatever  it  loses  by  the  diminished  weight  be- 
ing gained  by  the  increased  leverage.  An  axle,  the  surface 
of  which  is  curved  in  such  a  manner,  that  the  thickness  on 


154  THE    ELEMENTS    OP    MECHANICS.  CHAP.    XIV. 

which  the  rope  is  coiled  continually  increases  or  diminishes 
in  the  same  proportion  as  the  weight  or  resistance  diminishes 
or  increases,  will  produce  this  effect. 

It  is  obvious  that  all  that  has  been  said  respecting  a  variable 
weight  or  resistance,  is  also  applicable  to  a  variable  power, 
which,  therefore,  may,  by  the  same  means,  be  made  to  pro- 
duce a  uniform  effect.  An  instance  of  this  occurs  in  a  watch, 
which  is  moved  by  a  spiral  spring.  When  the  watch  has 
been  wound  up,  this  spring  acts  with  its  greatest  intensity, 
and,  as  the  watch  goes  down,  the  elastic  force  of  the  spring 
gradually  loses  its  energy.  This  spring  is  connected  by  a 
chain  with  an  axle  of  varying  thickness,  called  a  fusee.  When 
the  spring  is  at  its"  greatest  intensity,  the  chain  acts  upon  the 
thinnest  part  of  the  fusee,  and  as  it  is  uncoiled,  it  acts  upon 
a  part  of  the  fusee  which  is  continually  increasing  in  thick- 
ness, the  spring  at  the  same  time  losing  its  elastic  power  in 
exactly  the  same  proportion.  A  representation  of  the  fusee, 
and  the  cylindrical  box  which  contains  the  spring,  is  given  in 
fig.  98.,  and  of  the  spring  itself  in  Jig.  99. 

(256.)  When  great  power  is  required,  wheels  and  axles 
may  be  combined  in  a  manner  analogous  to  a  compound  sys- 
tem of  levers,  explained  in  (246.)  In  this  case  the  power 
acts  on  the  circumference  of  the  first  wheel,  and  its  effect  is 
transmitted  to  the  circumference  of  the  first  axle.  That  cir- 
cumference is  placed  in  connection  with  the  circumference  of 
the  second  wheel,  and  the  effect  is  thereby  transmitted  to  the 
circumference  of  the  second  axle,  and  so  on.  It  is  obvious 
from  what  was  proved  in  (248.),  that  the  power  of  such  a 
combination  of  wheels  and  axles  will  be  found  by  multiplying 
together  the  powers  of  the  several  wheels  of  which  it  is  com- 
posed. It  is  sometimes  convenient  to  compute  this  power  by 
numbers,  expressing  the  proportions  of  the  circumferences  or 
diameters  of  the  several  wheels,  to  the  circumferences  or  di- 
ameters of  the  several  axles  respectively.  This  computation 
is  made  by  first  multiplying  the  numbers  together  which  ex- 
press the  circumferences  or  diameters  of  the  wheels,  and  then 
multiplying  together  the  numbers  which  express  the  circum- 
ferences or  diameters  of  the  several  axles.  The  proportion 
of  the  two  products  will  express  the  power  of  the  machine. 
Thus,  if  the  circumferences  or  diameters  be  as  the  numbers 
10,  14,  and  15,  their  product  will  be  2100 ;  and  if  the  cir- 
cumferences or  diameters  of  the  axles  be  expressed  by  the 
numbers  3,  4,  and  5,  their  product  will  be  60,  and  the  power 


CHAP.    XIV.  COMPOUND    WHEEL-WORK.  155 

of  the  machine  will  be  expressed  by  the  proportion  of  2100 
and  60,  or  35  to  1. 

(257.)  The  manner  in  which  the  circumferences  of  the 
axles  act  upon  the  circumferences  of  the  wheels  in  com- 
pound wheel-work  is  various.  Sometimes  a  strap  or  cord  is 
applied  to  a  groove  in  the  circumference  of  the  axle,  and 
carried  round  a  similar  groove  in  the  circumference  of  the 
succeeding  wheel.  The  friction  of  this  cord  or  strap  with 
the  groove  is  sufficient  to  prevent  its  sliding,  and  to  commu- 
nicate the  force  from  the  axle  to  the  wheel,  or  vice  versd. 
This  method  of  connecting  wheel-work  is  represented  in 

fig- 100. 

Numerous  examples  of  wheels  and  axles  driven  by  straps 
or  cords  occur  in  machinery,  applied  to  almost  every  depart- 
ment of  the  arts  and  manufactures.  In  the  turning  lathe, 
the  wheel  worked  by  the  trcddle  is  connected  with  the  man- 
drel by  a  catgut  cord  passing  through  grooves  in  the  wheel 
and  axle.  In  all  great  factories,  revolving  shafts  are  carried 
along  the  apartments,  on  which,  at  certain  intervals,  straps 
are  attached,  passing  round  their  circumferences,  and  carried 
round  the  wheels  which  give  motion  to  the  several  machines. 
If  the  wheels,  connected  by  straps  or  cords,  are  required  to 
revolve  in  the  same  direction,  these  cords  are  arranged  as  in 
Jig.  100.  ;  but  if  they  are  required  to  revolve  in  contrary 
directions,  they  are  applied  as  in  Jig.  101. 

One  of  the  chief  advantages  of  the  method  of  transmitting 
motion  between  wheels  and  axles  by  straps  or  cords,  is,  that 
the  wheel  and  axle  may  be  placed  at  any  distance  from  each 
other  which  may  be  found  convenient,  and  may  be  made  to 
turn  either  in  the  same  or  contrary  directions. 

(258.)  When  the  circumference  of  the  wheel  acts  imme- 
diately on  the  circumference  of  the  succeeding  axle,  some 
means  must  necessarily  be  adopted  to  prevent  the  wheel  from 
moving  in  contact  with  the  axle  without  compelling  the  latter 
to  turn.  If  the  surfaces  of  both  were  perfectly  smooth,  so 
that  all  friction  were  removed,  it  is  obvious  that  either  would 
slide  over  the  surface  of  the  other,  without  communicating 
motion  to  it.  But,  on  the  other  hand,  if  there  were  any  as- 
perities, however  small,  upon  these  surfaces,  they  would 
become  mutually  inserted  among  each  other,  and  neither  the 
wheel  nor  axle  could  move  without  causing  the  asperities 
with  which  its  edge  is  studded  to  encounter  those  asperities 
which  project  from  the  surface  of  the  other  :  and  thus  until 


156  THE    ELEMENTS    OF    MECHANICS.  CHAP.    XIV 

these  projections  should  be  broken  off,  both  wheel  and  axle 
must  be  moved  at  the  same  time.  It  is  on  this  account  that, 
if  the  surfaces  of  the  wheels  and  axles  are  by  any  means 
rendered  rough,  and  pressed  together  with  sufficient  force,  the 
motion  of  either  will  turn  the  other,  provided  the  load  or 
resistance  be  not  greater  than  the  force  necesssary  to  break 
off  these  small  projections  which  produce  the  friction. 

In  cases  where  great  power  is  not  required,  motion  is  com- 
municated in  this  way  through  a  train  of  wheel-work,  by 
rendering  the  surface  of  the  wheel  and  axle  rough,  either  by 
facing  them  with  buff  leather,  or  with  wood  cut  across  the 
grain.  This  method  is  sometimes  used  in  spinning  ma- 
chinery, where  one  large  buffed  wheel,  placed  in  a  horizontal 
position,  revolves  in  contact  with  several  small  buffed  rollers, 
each  roller  communicating  motion  to  a  spindle.  The  position 
of  the  wheel  W,  and  the  rollers  11  R,  &c.,  are  represented 
in  Jig.  102.  Each  roller  can  be  thrown  out  of  contact  with 
the  wheel,  and  restored  to  it  at  pleasure. 

The  communication  of  motion  between  wheels  and  axles 
by  friction  has  the  advantage  of  great  smoothness  and  even- 
ness, and  of  proceeding  with  little  noise  ;  but  this  method 
can  only  be  used  in  cases  where  the  resistance  is  not  very 
considerable,  and,  therefore,  is  seldom  adopted  in  works  on  a 
large  scale.  Dr.  Gregory  mentions  an  instance  of  a  saw-mill 
at  Southampton,  where  the  wheels  act  upon  each  other  by 
the  contact  of  the  end  grain  of  wood.  The  machinery 
makes  very  little  noise,  and  wears  very  well,  having  been 
used  not  less  than  20  years. 

(259.)  The  most  nisual  method  of  transmitting  motion 
through  a  train  of  wheel-work  is  by  the  formation  of  teeth 
upon  their  circumferences,  so  that  these  indentures  of  each 
wheel  fall  between  the  corresponding  ones  of  that  in  which 
it  works,  and  ensure  the  action  so  long  as  the  strain  is  not 
so  great  as  to  fracture  the  tooth. 

In  the  formation  of  teeth,  very  minute  attention  must  be 
given  to  their  figure,  in  order  that  the  motion  may  be  com- 
municated from  wheel  to  wheel  with  smoothness  and  uni- 
formity. This  can  only  be  accomplished  by  shaping  the 
teeth  according  to  curves  of  a  peculiar  kind,  which  mathe- 
maticians have  invented,  and  assigned  rules  for  drawing. 
The  ill  consequences  of  neglecting  this  will  be  very  apparent, 
by  considering  the  nature  of  the  action  which  would  be  pro- 
duced if  the  teeth  were  formed  of  square  projecting  pins,  as 


C1IAI*.  XIV.  TOOTHED    WHEELS.  157 

m  fig>  103.  When  the  tooth  A  comes  into  contact  with  B, 
it  acts  obliquely  upon  it,  and,  as  it  moves,  the  corner  of  B 
slides  upon  the  plane  surface  of  A  in  such  a  manner  as  to 
produce  much  friction,  and  to  grind  away  the  side  of  A  and 
the  end  of  B.  As  they  approach  the  position  CD,  they  sus- 
tain a  jolt  the  moment  their  surfaces  come  into  full  contact ; 
and  after  passing  the  position  of  C  D,  the  same  scraping  and 
grinding  effect  is  produced  in  the  opposite  direction,  until, 
by  the  revolution  of  the  wheels,  the  teeth  become  disengaged. 
These  effects  are  avoided  by  giving  to  the  teeth  the  curved 
forms  represented  in, fig-  104.  By  such  means  the  surfaces 
of  the  teeth  roll  upon  each  other  with  very  inconsiderable 
friction,  and  the  direction  in  which  the  pressure  is  excited  is 
always  that  of  a  line  M  N,  touching  the  two  wheels,  and  at 
right  angles  to  the  radii.  Thus  the  pressure,  being  always 
the  same,  and  acting  with  the  same  leverage,  produces  a 
uniform  effect. 

(260.)  When  wheels  work  together,  their  teeth  must 
necessarily  be  the  same  size,  and  therefore  the  proportion  of 
their  circumferences  may  always  be  estimated  by  the  number 
of  teeth  which  they  carry.  Hence  it  follows,  that  in  com- 
puting the  power  of  compound  wheel-work,  the  number  of 
teeth  may  always  be  used  to  express  the  circumferences 
respectively,  or  the  diameters  which  are  proportional  to  these 
circumferences.  When  teeth  are  raised  upon  an  axle,  it  is 
generally  called  a  pinion,  and  in  that  case  the  teeth  are  called 
leaves.  The  rule  for  computing  the  train  of  wheel-work  given 
in  (256.)  will  be  expressed  as  follows :  when  the  wheel  and 
axle  carry  teeth,  multiply  together  the  number  of  teeth  in 
each  of  the  wheels,  and  next  the  number  of  leaves  in  each 
of  the  pinions  ;  the  proportion  of  the  two  products  will  ex- 
press the  power  of  the  machine.  If  some  of  the  wheels  and 
axles  carry  teeth,  and  others  not,  this  computation  may  be 
made  by  using  for  those  circumferences  which  do  not  bear 
teeth  the  number  of  teeth  which  would  fill  them.  Fig.  105. 
represents  a  train  of  three  wheels  and  pinions.  The  wheel 
F  which  bears  the  power,  and  the  axle  which  bears  the 
weight,  have  no  teeth  :  but  it  is  easy  to  find  the  number  of 
teeth  which  they  would  carry. 

(261.)  It  is  evident  that  each  pinion  revolves  much  more 

frequently  in  a  given  time  than  the  wheel  which  it  drives. 

Thus,  if  the  pinion  C  be  furnished  with  ten  teeth,  and  the 

wheel  E,  which  it  drives,  have  sixty  teeth,  the  pinion  C  must 

14 


158  THE    ELEMENTS    OF    MECHANICS.  CHAP.    XIV 

turn  six  times,  in  order  to  turn  the  wheel  E  once  round. 
The  velocities  of  revolution  of  every  wheel  and  pinion  which 
work  in  one  another,  will,  therefore,  have  the  same  proportion 
as  their  number  of  teeth  taken  in  a  reverse  order,  and  by  this 
means  the  relative  velocity  of  wheels  and  pinions  may  be  de- 
termined according  to  any  proposed  rate. 

Wheel-work,  like  all  other  machinery,  is  used  to  transmit 
and  modify  force  in  every  department  of  the  arts  and  manu- 
factures ;  but  it  is  also  used  in  cases  where  motion  alone,  and 
not  force,  is  the  object  to  be  attained.  The  most  remarkable 
example  of  this  occurs  in  watch  and  clock-work,  where  the 
object  is  merely  to  produce  uniform  motions  of  rotation, 
having  certain  proportions,  and  without  any  regard  to  the 
elevation  of  weights,  or  the  overcoming  of  resistances. 

(262.)  A  crane  is  an  example  of  combination  of  wheel- 
work  used  for  the  purpose  of  raising  or  lowering  great 
weights.  Fig.  106.  represents  a  machine  of  this  kind. 
A  B  is  a  strong  vertical  beam,  resting  on  a  pivot,  and  secur- 
ed in  its  position  by  beams  in  the  floor.  It  is  capable, 
however,  of  turning  on  its  axis,  being  confined  between 
rollers  attached  to  the  beams  and  fixed  in  the  floor.  C  D  is 
a  projecting  arm  called  a  gib,  formed  of  beams  which  are 
mortised  into  A  B.  The  wheel-work  is  mounted  in  two  cast- 
iron  crosses,  bolted  on  each  side  of  the  beams,  one  of  which 
appears  at  E  F  G  H.  The  winch  at  which  the  power  is  ap- 
plied is  at  I.  This  carries  a  pinion  immediately  behind  H. 
This  pinion  works  in  a  wheel  K,  which  carries  another 
pinion  upon  its  axle.  This  last  pinion  works  in  a  larger 
wheel  L,  which  carries  upon  its  axis  a  barrel  M,  on  which  a 
chain  or  rope  is  coiled.  The  chain  passes  over  a  pulley  D 
at  the  top  of  the  gib.  At  the  end  of  the  chain  a  hook 
O  is  attached,  to  support  the  weight  W.  During  the  eleva- 
tion of  the  weight,  it  is  convenient  that  its  recoil  should  be 
hindered  in  case  of  any  occasional  suspension  of  the  power. 
This  is  accomplished  by  a  ratchet  wheel  attached  to  the  bar- 
rel M,  as  explained  in  (253.) ;  but  when  the  weight  W  is  to 
be  lowered,  the  catch  must  be  removed  from  this  ratchet 
wheel.  In  this  case,  the  too  rapid  descent  of  the  weight  is 
in  some  cases  checked  by  pressure  excited  on  some  part  of 
the  wheel-work,  so  as  to  produce  sufficient  friction  to  retard 
the  descent  in  any  required  degree,  or  even  to  suspend  it,  if 
necessary.  The  vertical  beam  at  B  resting  on  a  pivot,  and 
being  fixed  between  rollers,  allows  the  gib  to  be  turned  round 


CHAP.  XIV.  REVELLED  GEAR.  159 

in  any  direction ;  so  that  a  weight  raised  from  one  side  of 
the  crane  may  be  carried  round,  and  deposited  on  another 
side,  at  any  distance  within  the  range  of  the  gib.  Thus,  if 
a  crane  be  placed  upon  a  wharf  near  a  vessel,  weights  may 
be  raised,  and,  when  elevated,  the  gib  may  be  turned  round 
so  as  to  let  them  descend  into  the  hold. 

The  power  of  this  machine  may  be  computed  upon  the 
principles  already  explained.  The  magnitude  of  the  circle, 
in  which  the  power  at  I  moves,  may  be  determined  by  the  ra- 
dius of  the  winch,  and  therefore  the  number  of  teeth  which  a 
wheel  of  that  size  would  carry  may  be  found.  In  like  man- 
ner, we  may  determine  the  number  of  leaves  in  a  pinion 
whose  magnitude  would  be  equal  to  the  barrel  M.  Let  the 
first  number  be  multiplied  by  the  number  of  teeth  in  the 
wheel  K,  and  that  product  by  the  number  of  teeth  in  the 
wheel  L.  Next,  let  the  number  of  leaves  in  the  pinion  H  be 
multiplied  by  the  number  of  leaves  in  the  pinion  attached  to 
the  axle  of  the  wheel  K,  and  let  that  product  be  multiplied 
by  the  number  of  leaves  in  a  pinion  whose  diameter  is  equal 
to  that  of  the  barrel  M.  These  two  products  will  express  the 
power  of  the  machine. 

(263.)  Toothed  wheels  are  of  three  kinds,  distinguished 
by  the  position  which  the  teeth  bear  with  respect  to  the  axis 
of  the  wheel.  When  they  are  raised  upon  the  edge  of  the 
wheel  as  in  Jig.  105.,  they  are  called  spur  wheels  or  spur  gear. 
When  they  are  raised  parallel  to  the  axis,  as  in  Jig.  107.,  it 
is  called  a  crown  wheel.  When  the  teeth  are  raised  on  a  sur- 
face inclined  to  the  plane  of  the  wheel,  as  \njig.  108.,  they 
are  called  bevelled  wheels. 

If  a  motion  round  one  axis  is  to  be  communicated  to 
another  axis  parallel  to  it,  spur  gear  is  generally  used. 
Thus  in  Jig.  105.,  the  three  axes  are  parallel  to  each  other. 
If  a- motion  round  one  axis  is  to  be  communicated  to  another 
at  right  angles  to  it,  a  crown  wheel,  working  in  a  spur  pinion, 
as  in  Jig.  107.,  will  serve.  Or  the  same  object  may  be  ob- 
tained by  two  bevelled  wheels,  as  in  Jig.  108. 

If  a  motion  round  one  axis  is  required  to  be  communicated 
to  another  inclined  to  it  at  any  proposed  angle,  two  bevelled 
wheels  can  always  be  used.  \\\.jig.  109.,  let  A  B  and  A  C 
be  the  two  axles  ;  two  bevelled  wheels,  such  as  D  E  and  E  F, 
on  these  axles  will  transmit  the  motion  or  rotation  from  one 
to  the  other,  and  the  relative  velocity  may,  as  usual,  be  regu- 
lated by  the  proportional  magnitude  of  the  wheels. 


160  THE  ELEMENTS  OF  MECHANICS.     CHAP.  XIV. 

(264.)  In  order  to  equalize  the  wear  of  the  teeth  of  a 
wheel  and  pinion,  which  work  in  one  another,  it  is  necessary 
that  every  leaf  of  the  pinion  should  work  in  succession 
through  every  tooth  of  the  wheel,  and  not  continually  act 
upon  the  same  set  of  teeth.  If  the  teeth  could  be  accurately 
shaped  according  to  mathematical  principles,  and  the  mate- 
rials of  which  they  are  formed  be  perfectly  uniform,  this 
precaution  would  be  less  necessary  ;  but  as  slight  inequalities, 
both  of  material  and  form,  must  necessarily  exist,  the  effects 
of  these  should  be  as  far  as  possible  equalized,  by  distributing 
them  through  every  part  of  the  wheel.  For  this  purpose,  it 
is  usual,  especially  in  mill-work,  where  considerable  force  is 
used,  so  to  regulate  the  proportion  of  the  number  of  teeth  in 
the  wheel  and  pinion,  that  the  same  leaf  of  the  pinion  shall 
not  be  engaged  twice  with  any  one  tooth  of  the  wheel,  until 
after  the  action  of  a  number  of  teeth,  expressed  by  the  prod- 
uct of  the  number  of  teeth  in  the  wheel  and  pinion.  Let 
us  suppose  that  the  pinion  contains  ten  leaves,  which  we 
shall  denominate  by  the  numbers  1,  2,  3,  &,c.,  and  that  the 
wheel  contains  60  teeth  similarly  denominated.  At  the 
commencement  of  the  motion,  suppose  the  leaf  1  of  the  pin- 
ion engages  the  tooth  1  of  the  wheel;  then  after  one. revolu- 
tion the  leaf  1  of  the  pinion  will  engage  the  tooth  11  of  the 
wheel,  and  after  two  revolutions  the  leaf  1  of  the  pinion  will 
engage  the  tooth  21  of  the  wheel,  and  in  like  manner,  after 
8,  4,  and  5  revolutions  of  the  pinion,  the  leaf  1  will  engage 
successively  the  teeth  31,  41,  and  51  of  the  wheel.  After  the 
sixth  revolution,  the  leaf  1  of  the  pinion  will  engage  the 
tooth  1  of  the  wheel.  Thus  it  is  evident,  that,  in  the  case 
here  supposed,  the  leaf  1  of  the  pinion  will  continually  be 
engaged  with  the  teeth  1,11,  21,  31,  41,  and  51  of  the 
wheel,  and  no  others.  The  like  may  be  said  of  every  leaf 
of  the  pinion.  Thus  the  leaf  2  of  the  pinion  will  bo  succes- 
ively  engaged  with  the  teeth  2,  12,  22,  32,  42,  and  52  of 
the  wheel,  and  no  others.  Any  accidental  inequalities  of 
these  teeth  will  therefore  continually  act  upon  each  other, 
until  the  circumference  of  the  wheel  be  divided  into  parts  of 
ten  teeth  each,  unequally  worn.  This  eifect  would  be 
avoided  by  giving  either  the  wheel  or  pinion  one  tooth  more 
or  one  tooth  less.  Thus,  suppose  the  wheel,  instead  of  hav- 
ing sixty  teeth,  had  sixty-one,  then  after  six  revolutions  of 
the  pinion  the  leaf  1  of  the  pinion  would  be  engaged  with 
the  tooth  61  of  the  wheel :  and  after  one  revolution  of  th« 


CHAP.    XIV.  WATCH    AND    CLOCK-WORK.  161 

wheel,  the  leaf  2  of  the  pinion  would  he  engaged  with  the 
tooth  1  of  the  wheel.  Thus,  during  the  first  revolution  of 
the  wheel,  the  leaf  1  of  the  pinion  would  he  successively  en- 
gaged with  the  teeth  1,  11,  21,  31,  41,  51,  and  61  of  the 
wheel ;  at  the  commencement  of  the  second  revolution  of 
the  wheel  the  leaf  2  of  the  pinion  would  be  engaged  with 
the  tooth  1  of  the  wheel  ;  and  during  the  second  revolution 
of  the  wheel  the  leaf  1  of  the  pinion  would  be  successively 
engaged  with  the  teeth  10,20,30,40,50,  and  60  of  the 
wheel.  In  the  same  manner  it  may  be  shown,  that  in  the 
third  revolution  of  the  wheel  the  leaf  1  of  the  pinion  would 
be  successively  engaged  with  the  teeth  9,  19,  29,  39,  49,  and 
59  of  the  wheel  ;  during  the  fourth  revolution  of  the  wheel, 
the  leaf  1  of  the  pinion  would  be  successively  engaged  with 
the  teeth, 8,  18,  28,  38,  48,  and  58  of  the  wheel.  By  con- 
tinuing this  reasoning  it  will  appear,  that  during  the  tenth 
revolution  of  the  wheel  the  leaf  1  of  the  pinion  will  be  en- 
gaged successively  with  the  teeth  2,  12,  22,  32,  42,  and  52 
of  the  wheel.  At  the  commencement  of  the  eleventh  revo- 
lution of  the  wheel  the  leaf  1  of  the  pinion  will  be  engaged 
with  the  tooth  1  of  the  wheel,  as  at  the  beginning  of  the 
motion.  It  is  evident,  therefore,  that  during  the  first  ten 
revolutions  of  the  wheel  each  leaf  of  the  pinion  has  been 
successively  engaged  with  every  tooth  of  the  wheel,  and  that 
during  these  ten  revolutions  the  pinion  has  revolved  sixty-one 
times.  Thus  the  leaves  of  the  pinion  have  acted  six  hun- 
dred and  ten  times  upon  the  teeth  of  the  wheel,  before  two 
teeth  can  have  acted  twice  upon  each  other. 

The  odd  tooth  which  produces  this  effect  is  called  by  mill- 
wrights the  hunting  cog. 

(265.)  The  most  familiar  case  in  which  wheel-work  is 
used  to  produce  and  regulate  motion  merely,  without  any 
reference  to  weights  to  be  raised  or  resistances  to  be  over- 
come, is  that  of  chronometers.  In  watch  and  clock-work, 
the  object  is  to  cause  a  wheel  to  revolve  with  a  uniform  ve- 
locity, and  at  a  certain  rate.  The  motion  of  this  wheel  is 
indicated  by  an  index  or  hand  placed  upon  its  axis,  and 
carried  round  with  it.  In  proportion  to  the  length  of  the 
hand,  the  circle  over  which  its  extremity  plays  is  enlarged, 
and  its  motion  becomes  more  perceptible.  This  circle 
is  divided,  so  that  very  small  fractions  of  a  revolution  of  the 
hand  may  be  accurately  observed.  In  most  chronometers,  it 
is  required  to  give  motion  to  two  hands,  and  sometimes  to 
14  * 


THE  ELEMENTS  OF  MECHANICS.     CHAP.  XIV. 

three.  These  motions  proceed  at  different  rates,  according 
to  the  subdivisions  of  time  generally  adopted.  One  wheel 
revolves  in  a  minute,  hearing  a  hand  which  plays  round  a 
circle  divided  into  sixty  equal  parts;  the  motion  of  the  hand 
over  each  part  indicating  one  second,  and  a  complete  revolu- 
tion of  the  hand  being  performed  in  one  minute.  Another 
wheel  revolves  once,  while  the  former  revolves  sixty  times ; 
consequently  the  hand  carried  by  this  wheel  revolves  once  in 
sixty  minutes,  or  one  hour.  The  circle  on  which  it  plays  is 
like  the  former,  divided  into  sixty  equal  parts,  and  the  motion 
of  the  hand  over  each  division  is  performed  in  one  minute. 
This  is  generally  called  the  minute,  hand,  and  the  former  the 
second  hand. 

A  third  wheel  revolves  once,  while  that  which  carries  the 
minute  hand  revolves  twelve  times;  consequently  this  last 
wheel,  which  carries  the  hour  hand,  revolves  at  a  rate  twelve 
times  less  than  that  of  the  minute  hand,  and  therefore  seven 
hundred  and  twenty  times  less  than  the  second  hand.  We 
shall  now  endeavor  to  explain  the  manner  in  which  these 
motions  are  produced  and  regulated.  Let  A,  B,  C,  D,  E, 
fig.  110.,  represent  a  train  of  wheels,  and  «,  b,  r,  d,  repre- 
sent their  pinions,  e  being  a  cylinder  on  the  axis  of  the  wheel 
E,  round  which  a  rope  is  coiled,  sustaining  a  weight  W. 
Let  the  effect  of  this  weight,  transmitted  through  the  train 
of  wheels,  be  opposed  by  a  power  P  acting  upon  the  wheel  A, 
and  let  this  power  be  supposed  to  be  of  such  a  nature  as  to 
cause  the  weight  W  to  descend  with  a  uniform  velocity,  and 
at  any  proposed  rate.  The  wheel  E  carries  on  its  circumfer- 
ence eighty-four  teeth.  The  wheel  D  carries  eighty  teeth ; 
the  wheel  C  is  also  furnished  with  eighty  teeth,  and  the 
wheel  B  with  seventy-five.  The  pinions  d  and  c  are  each  fur- 
nished with  twelve  leaves,  and  the  pinions  b  and  a  with  ten. 

If  the  power  at  P  be  so  regulated  as  to  allow  the  wheel  A 
to  revolve  once  in  a  minute,  with  a  uniform  velocity,  a  hami 
attached  to  the  axis  of  this  wheel  will  serve  as  the  second 
hand.  The  pinion  a  carrying  ten  teeth  must  revolve  seven 
times  and  a  half  to  produce  one  revolution  of  B,  consequent- 
ly fifteen  revolutions  of  the  wheel  A  will  produce  two  revolu- 
tions of  the  wheel  B  ;  the  wheel  B,  therefore,  revolves  twice 
in  fifteen  minutes.  The  pinion  b  must  revolve  eight  times 
to  produce  one  revolution  of  the  wheel  C>  and  therefore  the 
wheel  C  must  revolve  once  in  four  quarters  of  an  hour,  or 
in  one  hour.  If  a  hand  be  attached  to  the  axis  of  this 


CHAP.  XIV.  CLOCK-WORK PENDULUM.  163 

wheel,  it  will  have  the  motion  necessary  for  the  minute  hand. 
The  pinion  c  must  revolve  six  times  and  two  thirds  to  produce 
one  revolution  of  the  wheel  D,  and  therefore  this  wheel  must 
revolve  once  in  six  hours  and  two  thirds.  The  pinion  d 
revolves  seven  times  for  one  revolution  of  the  wheel  E,  and 
therefore  the  wheel  E  will  revolve  once  in  forty-six  hours  and 
two  thirds. 

On  the  axis  of  the  wheel  C  a  second  pinion  may  be  placed, 
furnished  with  seven  leaves,  which  may  lead  a  wheel  of  eighty- 
four  teeth,  so  that  this  wheel  shall  turn  once  during  twelve 
turns  of  the  wheel  C.  If  a  hand  be  fixed  upon  the  axis, 
this  hand  will  revolve  once  for  twelve  revolutions  of  the  min- 
ute hand  fixed  upon  the  axis  of  the  wheel- C  ;  that  is,  it  will 
revolve  once  in  twelve  hours.  If  it  play  upon  a  dial  divi- 
ded into  twelve  equal  parts,  it  will  move  over  each  part  in  an 
lur.ir,  ;i:i<l  will  serve  the  purpose  of  the  hour  hand  of  the 
chronometer. 

We  have  here  supposed  that  the  second  hand,  the  minute 
hand  and  the  hour  hand  move  on  separate  dials.  This,  how- 
ever, is  not  necessary.  The  axis  of  the  hour  hand  is  com- 
monly a  tube,  enclosing  within  it  that  of  the  minute  hand,  so 
that  the  same  dial  serves  for  both.  The  second  hand,  how- 
ever, is  generally  furnished  with  a  separate  dial. 

(266.)  We  shall  now  explain  the  manner  in  which  a  power 
is  applied  to  the  wheel  A,  so  as  to  regulate  and  equalize  the 
effect  of  the  weight  W.  Suppose  the  wheel  A  furnished 
with  thirty  teeth,  as  in  Jig.  111.;  if  nothing  check  the  mo- 
tion, the  weight  W  would  descend  with  an  accelerated  velocity, 
and  would  communicate  an  accelerated  motion  to  the  wheel 
A.  This  effect,  however,  is  interrupted  by  the  following 
contrivance  : — L  M  is  a  pendulum  vibrating  on  the  centre  L, 
and  so  regulated  that  the  time  of  its  oscillation  is  one  second. 
The  pallets  1  and  K  are  connected  with  the  pendulum,  so  as 
to  oscillate  with  it.  In  the  position  of  the  pendulum  repre- 
sented in  the  figure,  the  pallet  I  stops  the  motion  of  the  wheel 
A,  and  entirely  suspends  the  action  of  the  weight  W ,Jig.  110., 
KO  that  for  a  moment  the  entire  machine  is  motionless.  The 
weight  M,  however,  falls  by  its  gravity  towards  tiie  lowest 
position,  and  disengages  the  pallet  I  from  the  tooth  of  the 
wheel.  The  weight  W  begins  then  to  take  effect,  and  the 
wheel  A  turns  from  A  towards  B.  Meanwhile  the  pendulum 
M  oscillates  to  the  other  side,  and  the  pallet  K  falls  under 
a  tooth  of  the  wheel  A,  and  checks  for  a  moment  its  further 


164  THE  ELEMENTS  OF  MECHANICS.     CHAP.  XIV. 

motion.  On  the  returning  vibration,  the  pallet  K  becomes 
again  disengaged,  and  allows  the  tooth  of  the  wheel  to  escape, 
and  by  the  influence  of  the  weight  W  another  tooth  passes 
before  the  motion  of  the  wheel  A  is  again  checked  by  the 
interposition  of  the  pallet  I. 

From  this  explanation  it  will  appear  that,  in  two  vibrations 
of  the  pendulum,  one  tooth  of  the  wheel  A  passes  the  pallet 
I,  and,  therefore,  if  the  wheel  A  be  furnished  with  30  teeth, 
it  will  be  allowed  to  make  one  revolution  during  60  vibrations 
of  the  pendulum.  If,  therefore,  the  pendulum  be  regulated 
so  as  to  vibrate  seconds,  this  wheel  will  revolve  once  in  a 
minute.  From  the  action  of  the  pallets  in  checking  the  mo- 
tion of  the  wheel  A,  and  allowing  its  leeth  alternately  to 
escape,  -this  has  been  called  the  cur  ape  went  wheel  ;  and  the 
wheel  and  pallets  together  are  generally  called  the  csccipcment 
or  'scapement. 

We  have  already  explained,  that  by  reason  of  the  frictioa 
on  the  points  of  support,  and  other  causes,  the  swing  of  the 
pendulum  would  gradually  diminish,  and  its  vibration  at  length 
cease.  This,  however,  is  prevented  by  the  action  of  the 
teeth  of  the  scapement  wheel  upon  the  pallets,  which  is  just 
sufficient  to  communicate  that  quantity  of  force  to  the  pen- 
dulum which  is  necessary  to  counteract  the  retarding  effects, 
and  to  maintain  its  motion.  It  thus  appears,  that  although 
the  effect  of  the  gravity  of  the  weight  W  in  giving  motion  to 
the  machine  is  at  intervals  suspended,  yet  this  part  of  the 
force  is  not  lost,  being,  during  these  intervals,  employed  in 
giving  to  the  pendulum  all  that  motion  which  it  would  lose 
by  the  resistances  to  which  it  is  inevitably  exposed. 

In  stationary  clocks,  and  in  other  cases  in  which  the  bulk 
of  the  machine  is  not  au  objection,  a  descending  weight  is 
used  as  the  moving  power.  But  in  watches  and  portable 
chronometers,  this  would  be  attended  with  evident  inconve- 
nience. In  such  cases,  a  spiral  spring,  called  the  main  spring, 
is  the  moving  power.  The  manner  in  which  this  spring 
communicates  rotation  to  an  axis,  and  the  ingenious  method 
of  equalizing  the  effect  of  its  variable  elasticity  by  giving 
to  it  a  leverage,  which  increases  as  the  elastic  force  dimin- 
ishes, has  been  already  explained.  (255.) 

A  similar  objection  lies  against  the  use  of  a  pendulum  in 
portable  chronometers.  A  spiral  spring  of  a  similar  kind, 
but  infinitely  more  delicate,  called  a  hair  S2)ring,  is  substi- 
tuted in  its  place.  This  spring  is  connected  with  a  nicely 


CHAP.  XIV.  WATCH  AND  CLOCK-WORK.  165 

balanced  wheel,  called  the  balance  wheel)  which  plays  in  piv- 
ots. When  this  wheel  is  turned  to  a  certain  extent  in  one  di- 
rection, the  hair  spring  is  coiled  up,  and  its  elasticity  causes  the 
wheel  to  recoil,  and  return  to  a  position  in  which  the  energy 
of  the  spring  acts  in  the  opposite  direction.  The  balance 
wheel  then  returns,  and  continually  vibrates  in  the  same  man- 
ner. The  axis  of  this  wheel  is  furnished  with  pallets  similar 
to  those  of  the  pendulum,  which  are  alternately  engaged 
with  the  teeth  of  a  crown  wheel,  which  takes  the  place  of  the 
scapement  wheel  already  described. 

A  general  view  of  the  work  of  a  common  watch  is  repre- 
sented in  jig.  111.  bis.  A  is  the  balance  wheel  bearing 
pallets  p  p  upon  its  axis ;  C  is  the  crown  wheel,  whose  teeth 
are  suffered  to  escape  alternately  by  those  pallets  in  the  man- 
ner already  described  in  the  scapement  of  a  clock.  On  the 
axis  of  the  crown  wheel  is  placed  a  pinion  d,  which  drives 
another  crown  wheel  K.  On  the  axis  of  this  is  placed  the 
pinion  £,  which  plays  in  the  teeth  of  the  third  wheel  L.  The 
pinion  b  on  the  axis  of  L  is  engaged  with  the  wheel  M,  called 
the  centre  wheel.  The  axle  of  this  wheel  is  carried  up 
through  the  centre  of  the  dial.  A  pinion  a  is  placed  upon  it, 
which  works  in  the  great  wheel  N.  On  this  wheel  the  main 
spring  immediately  acts.  O  P  is  the  main  spring  stripped 
of  its  barrel.  The  axis  of  the  wheel  M  passing  through  the 
centre  of  the  dial  is  squared  at  the  end  to  receive  the  minute 
hand.  A  second  pinion  Q,  is  placed  upon  this  axle,  which 
drives  a  wheel  T.  On  the  axle  of  this  wheel  a  pinion  g 
is  placed,  which  drives  the  hour  wheel  V.  This  wheel  is 
placed  upon  a  tubular  axis,  which  encloses  within  it  the  axis 
of  the  wheel  M.  This  tubular  axis,  passing  through  the 
centre  of  the  dial,  carries  the  hour  hand.  The  wheels  A,  B, 
C,  D,  E,  Jig.  110.,  correspond  to  the  wheels  C,  K,  L,  M,  N, 
Jig.  112. ;  and  the  pinions  «,  b,  r,  e?,  p,  Jig.  109.,  correspond 
to  the  pinions  dy  r,  b,  a,  Jig.  111.  From  what  has  already 
been  explained  of  these  wheels,  it  will  be  obvious  that  the 
wheel  M,  jfc,  111.,  revolves  once  in  an  hour,  causing  the 
minute  hand  to  move  round  the  dial  once  in  that  time.  This 
wheel  at  the  same  time  turns  the  pinion  Q,  which  leads  the 
wheel  T.  This  wheel  again  turns  the  pinion  g,  which  leads 
the  hour  wheel  V.  The  leaves  and  teeth  of  these  pinions  and 
wheels  are  proportioned,  as  already  explained,  so  that  the 
wheel  V  revolves  once  during  twelve  revolutions  of  the  wheel 
M.  The  hour  hand,  therefore,  which  is  carried  by  the  tubu- 


166  THE  ELEMENTS  OF  MECHANICS.  CHAP.  XV. 

lar  axle  of  the  wheel  V,  moves  once  round  the  dial  in  twelve 
hours. 

Our  object  here  has  not  been  to  give  a  detailed  account 
of  watch  and  clock-work,  a  subject  for  which  we  must  refer 
the  reader  to  the  proper  department  of  this  work.  Such  a 
general  account  has  only  been  attempted  as  may  explain  how 
tooth  and  pinion  work  may  be  applied  to  regulate  motion. 


CHAPTER  XV. 

OP  THE  PULLEY. 

(267.)  THE  next  class  of  simple  machines,  which  present 
themselves  to  our  attention,  is  that  which  we  have  called  the 
cord.  If  a  rope  were  perfectly  flexible,  and  were  capable 
of  being  bent  over  a  sharp  edge,  and  of  moving  upon  it  with- 
out friction,  we  should  be  enabled  by  its  means  to  make  a 
force  in  any  one  direction  overcome  resistance,  or  communi- 
cate motion  in  any  other  direction.  Thus  if  P,  Jig.  112.,  be 
such  an  edge,  a  perfectly  flexible  rope  passing  over  it  would 
be  capable  of  transmitting  a  force  S  F  to  a  resistance  Q,  R, 
so  as  to  support  or  overcome  R,  or  by  a  motion  in  the  direc- 
tion of  S  F  to  produce  another  motion  in  the  direction  R  Q. 
But  as  no  materials  of  which  ropes  can  be  constructed  can 
give  them  perfect  flexibility,  and  as,  in  proportion  to  the 
strength  by  which  they  are  enabled  to  transmit  force,  their 
rigidity  increases,  it  is  necessary,  in  practice,  to  adopt  means 
to  remove  or  mitigate  those  effects  which  attend  imperfect 
flexibility,  and  which  would  otherwise  render  cords  practically 
inapplicable  as  machines. 

When  a  cord  is  used  to  transmit  a  force  from  one  direction 
to  another,  its  stiffness  renders  some  force  necessary  in  bend- 
ing it  over  the  angle  P,  which  the  two  directions  form  ;  and 
if  the  angle  be  sharp,  the  exertion  of  such  a  force  may  be 
attended  with  the  rupture  of  the  cord.  If,  instead  of  bending 
the  rope  at  one  point  over  a  single  angle,  the  change  of  direc- 
tion were  produced  by  successively  deflecting  it  over  several 
angles,  each  of  which  would  be  less  sharp  than  a  single  one 
could  be,  the  force  requisite  for  the  deflection,  as  well  as 
the  liability  of  rupturing  the  cord,  would  be  considerably 


CHAP.  XV.  FIXED  FULI.EIT.  167 

diminished.  But  this  end  will  be  still  more  perfectly  attained 
if  the  deflection  of  the  cord  be  produced  by  bending  it  over 
the  surface  of  a  curve. 

If  a  rope  were  applied  only  to  sustain,  and  not  to  move  a 
weight,  this  would  be  sufficient  to  remove  the  inconveniences 
arising  from  its  rigidity.  But  when  motion  is  to  be  produced, 
the  rope,  in  passing  over  the  curved  surface,  would  be  subject 
to  excessive  friction,  and  consequently  to  rapid  wear.  This 
inconvenience  is  removed  by  causing  the  surface  on  which 
the  rope  runs  to  move  with  it,  so  that  no  more  friction  is  pro- 
duced than  would  arise  from  the  curved  surface  rolling  upon 
the  rope. 

(268.)  All  these  ends  are  attained  by  the  common  pulley, 
which  consists  of  a  wheel  called  a  sheave,  fixed  in  a  block 
and  turning  on  pivots.  A  groove  is  formed  in  the  edge  of  the 
wheel,  in  which  the  rope  runs,  the  wheel  revolving  with  it 
Such  an  apparatus  is  represented  in  jig.  113, 

We  shall,  for  the  present,  omit  the  consideration  of  that 
part  of  the  effects  of  the  stiffness  and  friction  of  the  machine, 
which  is  not  removed  by  the  contrivance  just  explained,  and 
shall  consider  the  rope  as  perfectly  flexible,  and  moving  with- 
out friction. 

From  the  definition  of  a  flexible  cord,  it  follows,  that  its 
tension,  or  the  force  by  which  it  is  stretched  throughout  its 
entire  length,  must  be  uniform.  From  this  principle,  and 
this  alone,  all  the  mechanical  properties  of  pulleys  may  be  de- 
rived. 

Although,  as  already  explained,  the  whole  mechanical  effi- 
cacy of  this  machine  depends  on  the  qualities  of  the  cord, 
and  not  on  those  of  the  block  and  sheave,  which  are  only 
introduced  to  remove  the  accidental  effects  of  stiffness  and 
friction,  yet  it  has  been  usual  to  give  the  name  pulley  to 
the  block  and  sheave,  and  a  combination  of  blocks,  sheaves 
and  ropes  is  called  a  tackle. 

(269.)  When  the  rope  passes  over  a  single  wheel,  which 
is  fixed  in  its  position,  as  in  Jig.  113.,  the  machine  is  called 
a  fixed  pulley.  Since  the  tension  of  the  cord  is  uniform 
throughout  its  length,  it  follows,  that  *in  this  machine  the 
power  and  weight  are  equal.  For  the  weight  stretches  that 
part  of  the  cord  which  is  between  the  weight  and  pulley, 
and  the  power  stretches  that  part  between  the  power  and  the 
pulley.  And  since  the  tension  throughout  the  whole  length 
is  the  same,  the  weight  must  be  equal  to  the  power. 


168  THE  ELEMENTS  OF  MECHANICS.  CHAP.  XV. 

Hence  it  appears,  that  no  mechanical  advantage  is  gained 
by  this  machine.  Nevertheless,  there  is  scarcely  any  engine, 
simple  or  complex,  attended  with  more  convenience.  In  the 
application  of  power,  whether  of  men  or  animals,  or  arising 
from  natural  forces,  there  are  always  some  directions  in  which 
it  may  be  exerted  to  much  greater  convenience  and  advantage 
than  others,  and  in  many  cases  the  exertion  of  these  powers 
is  limited  to  a  single  direction.  A  machine,  therefore,  which 
enables  us  to  give  the  most  advantageous  direction  to  the 
moving  power,  whatever  be  the  direction  of  the  resistance 
opposed  to  it,  contributes  as  much  practical  convenience,  as 
one  which  enables  a  small  power  to  balance  or  overcome  a 
great  weight.  In  directing  the  power  against  the  resistance, 
it  is  often  necessary  to  use  two  fixed  pulleys.  Thus,  in  ele- 
vating a  weight  A,  Jig.  114.,  to  the  summit  of  a  building,  by 
the  strength  of  a  horse  moving  below,  two  fixed  pulleys,  B 
and  C,  may  be  used.  The  rope  is  carried  from  A  over  the 
pulley  B ;  the  rope  passes,  and,  returning  downwards,  is 
brought  under  C,  and  finally  drawn  by  the  animal  on  the 
horizontal  plane.  In  the  same  manner  sails  are  spread,  and 
flags  hoisted  on  the  yards  and  masts  of  a  ship,  by  sailors  pull- 
ing a  rope  on  the  deck. 

By  means  of  the  fixed  pulley  a  man  may  raise  himself  to  a 
considerable  height,  or  descend  to  any  proposed  depth.  If 
he  be  placed  in  a  chair  or  bucket  attached  to  one  end  of  a 
rope,  which  is  carried  over  a  fixed  pulley,  by  laying  hold  of 
this  rope  on  the  other  side,  as  represented  in  Jig-  115,,  he 
may,  at  will,  descend  to  a  depth  equal  to  half  of  the  entire 
length  of  the  rope,  by  continually  yielding  rope  on  the  one 
side,  and  depressing  the  bucket  or  chair  by  his  weight  on 
the  other.  Fire-escapes  have  been  constructed  on  this  prin- 
ciple, the  fixed  pulley  being  attached  to  some  part  of  the 
building. 

(270.)  A  single  movable  pulley  is  represented  in  Jig.  116. 
A  cord  is  carried  from  a  fixed  point  F,  and,  passing  through 
a  block  B,  attached  to  a  weight  W,  passes  over  a  fixed  pulley 
C,  the  power  being  applied  at  P.  We  shall  first  suppose  the 
parts  of  the  cord  on 'each  side  the  wheel  B  to  be  parallel ; 
in  this  case,  the  whole  weight  W  being  sustained  by  the 
parts  of  the  cords  B  C  and  B  F,  and  these  parts  being  equal- 
ly stretched  (268.),  each  must  sustain  half  the  weight,  which 
is  therefore  the  tension  of  the  cord.  This  tension  is  resisted 
by  the  power  at  P,  which  must,  therefore,  be  equal  to  hajf 


CHAP.  XV.  COMPOUND    PULLEYS.  169 

the  weight.     In  this  machine,  therefore,  the  weight  is  twice 
the  power. 

(271.)  If  the  parts  of  the  cord  B  C  and  B  F  be  not  paral- 
lel, as  in  Jig.  117.,  a  greater  power  than  half  the  weight 
is  therefore  necessary  to  sustain  it.  To  determine  the  power 
necessary  to  support  a  given  weight,  in  this  case  take  the  line 
B  A  in  the  vertical  direction,  consisting  of  as  many  inches 
as  the  weight  consists  of  ounces ;  from  A  draw  A  D  parallel 
to  B  C,  and  A  E  parallel  to  B  F ;  the  force  of  the  weight 
represented  by  A  B  will  be  equivalent  to  two  forces  repre- 
sented by  B  D  and  B  E.  (74.)  The  number  of  inches  in 
these  lines  respectively  will  represent  the  number  of  ounces 
which  are  equivalent  to  the  tensions  of  the  parts  B  F  and  B  C 
of  the  cord.  But  as  these  tensions  are  equal,  B  D  and  B  E 
must  be  equal,  and  each  will  express  the  amount  of  the  power 
P,  which  stretches  the  cord  at  P  C. 

It  is  evident  that  the  four  lines,  A  E,  E  B,  B  D,  and  D  A, 
are  equal.  And  as  each  of  them  represents  the  power,  the 
weight  which  is  represented  by  A  B  must  be  less  than  twice 
the  power  which  is  represented  by  A  E  and  E  B  taken 
together.  It  follows,  therefore,  that  as  parts  of  the  rope? 
which  support  the  weight  depart  from  parallelism,  the  machine 
becomes  less. and  less  efficacious;  and  there  are  certain  ob- 
liquities at  which  the  equilibrating  power  would  be  much 
greater  than  the  weight. 

(272.)  The  mechanical  power  of  pulleys  admits  of  being 
almost  indefinitely  increased  by  combination.  Systems  of 
pulleys  may  be  divided  into  two  classes ;  those  in  which  a 
single  rope  is  used,  and  those  which  consist  of  several  .dis- 
tinct ropes.  Figs.  118.  and  119.  represent  two  systems  of 
pulleys,  each  having  a  single  rope.  The  weight  is  in  each 
case  attached  to  a  movable  block  B,  in  which  are  fixed  two 
or  more  wheels ;  A  is  a  fixed  block,  and  the  rope  is  succes- 
sively passed  over  the  wheels  above  and  below,  and,  after 
passing  over  the  last  wheel  above,  is  attached  to  the  power. 
The  tension  of  that  part  of  the  cord  to  which  the  power  is 
attached  is  produced  by  the  power,  and  therefore  equivalent 
to  it,  and  the  same  tension  must  extend  throughout  its  whole 
length.  The  weight  is  sustained  by  all  those  parts  of  the  cord* 
which  pass  from  the  lower  block,  and,  as  the  force  which 
stretches  them  all  is  the  same,  viz.  that  of  the  power,  the  ef- 
fect of  the  weight  must  be  equally  distributed  among  them, 
their  directions  being  supposed  to  be  parallel.  It  will  be 
15 


170  THE    ELEMENTS    OF    MECHANICS.  CHAP.    XV. 

evident,  from  this  reasoning,  that  the  weight  will  be  as  many 
times  greater  than  the  power,  as  the  number  of  cords  which 
support  the  lower  block.  Thus,  if  there  be  six  cords,  each 
cord  will  support  a  sixth  part  of  the  weight,  that  is,  the  weight 
will  be  six  times  the  tension  of  the  cord,  or  six  times  the  pow- 
er. In  Jig.  118.  the  cord  is  represented  as  being  finally  at- 
tached to  a  hook  on  the  upper  block.  But  it  may  be  carried 
over  an  additional  wheel  fixed  in  that  block,  and  finally  at- 
tached to  a  hook  in  the  lower  block,  as  \njig.  119.,  by  which 
one  will  be  added  to  the  power  of  the  machine,  the  number 
of  cords  at  the  lower  block  being  increased  by  one.  In  the 
system  represented  in  Jig.  118.  the  wheels  are  placed  in  the 
blocks  one  above  the  other;  in  Jig.  119.  they  are  placed  side 
by  side.  In  all  systems  of  pulleys  of  this  class,  the  weight  of 
the  lower  block  is  to  be  considered  as  a  part  of  the  weight  to 
be  raised,  and,  in  estimating  the  power  of  the  machine,  this 
should  always  be  attended  to. 

(273.)  When  the  power  of  the  machine,  and  therefore  the 
number  of  wheels,  is  considerable,  some  difficulty  arises  in 
the  arrangement  of  the  wheels  and  cords.  The  celebrated 
Smeaton  contrived  a  tackle,  which  takes  its  name  from  him, 
in  which  there  are  ten  wheels  in  each  block  ;  five  large  wheels 
placed  side  by  side,  and  five  smaller  ones  similarly  placed 
above  them  in"  the  lower  block,  and  below  them  in  the  upper. 
Fig.  120.  represents  Smeaton's  blocks  without  the  rope.  The 
wheels  are  marked  with  the  numbers  1,  2,  3,  &.C.,  in  the  or- 
der in  which  the  rope  is  to  be  passed  over  them.  As  in  this 
pulley,  20  distinct  parts  of  the  rope  support  the  lower  block, 
the  weight,  including  the  lower  block,  will  be  20  times  the 
equilibrating  power. 

(274.)  In  all  these  systems  of  pulleys,  every  wheel  has  a 
separate  axle,  and  there  is  a  distinct  wheel  for  every  turn  of 
the  rope  at  each  block.  Each  wheel  is  attended  with  friction 
on  its  axle,  and  also  with  friction  between  the  sheave  and 
block.  The  machine  is  by  this  means  robbed  of  a  great  part 
of  its  efficacy,  since,  to  overcome  the  friction  alone,  a  consid- 
erable power  is  in  most  cases  necessary. 

An  ingenious  contrivance  has  been  suggested,  by  which  all 
the  advantage  of  a  large  number  of  wheels  may  be  obtained 
without  the  multiplied  friction  of  distinct  sheaves  and  axles. 
To  comprehend  the  excellence  of  this  contrivance,  it  will  be 
necessary  to  consider  the  rate  at  which  the  rope  passes  over 
the  several  wheels  of  such  a  system,  &sjig.  118.  If  one  foot 


CHAP.  xv.  WHITE'S  PULLEY.  171 

of  the  rope  G  F  pass  over  the  pulley  F,  two  feet  must  pass 
over  the  pulley  E,  because  the  distance  between  F  and  E 
being  shortened  one  foot,  the  total  length  of  the  rope  G  F  E 
must  be  shortened  two  feet.  These  two  feet  of  rope  must 
pass  in  the  direction  E  D,  and  the  wheel  D,  rising  one  foot, 
three  feet  of  rope  must  consequently  pass  over  it.  These  three 
feet  of  rope  passing  in  the  direction  D  C,  and  the  rope  D  C 
being  also  shortened  one  foot  by  the  ascent  of  the  lower  block, 
four  feet  of  rope  must  pass  over  the  wheel  C.  In  the  same 
way  it  may  be  shown  that  five  feet  must  pass  over  B,  and  six 
feet  over  A.  Thus,  whatever  be  the  number  of  wheels  in 
the  upper  and  lower  blocks,  the  parts  of  the  rope  which  pass 
in  the  same  time  over  the  wheels  in  the  lower  block  are  in 
the  proportion  of  the  odd  numbers  1,  3,  5,  &c. ;  and  those 
which  pass  over  the  wheels  in  the  upper  block  in  the  same 
time,  are  as  the  even  numbers  2,  4,  6,  &c.  If  the  wheels 
were  all  of  equal  size,  as  in  jig.  119.,  they  would  revolve  with 
velocities  proportional  to  the  rate  at  which  the  rope  passes 
over  them  :  so  that,  while  the  first  wheel  below  revolves  once, 
the  first  wheel  above  will  revolve  twice  :  the  second  wheel  be- 
low three  times ;  the  second  wheel  above  four  times,  and  so 
on.  If,  however,  the  wheels  differed  in  size  in  proportion  to 
the  quantity  of  rope  which  must  pass  over  them,  they  would 
evidently  revolve  in  the  same  time.  Thus,  if  the  first  wheel 
above  were  twice  the  size  of  the  first  wheel  below,  one  revo- 
lution would  throw  off  twice  the  quantity  of  rope.  Again,  if 
the  second  wheel  below  were  thrice  the  size  of  the  first  wheel 
below,  it  would  throw  off  in  one  revolution  thrice  the  quanti- 
ty of  rope,  and  so  on.  Wheels  thus  proportioned,  revolving 
in  exactly  the  same  time,  might  be  all  placed  on  one  axle, 
and  would  partake  of  one  common  motion,  or,  what  is  to  the 
same  effect,  several  grooves  might  be  cut  upon  the  face  of  one 
solid  wheel,  with  diameters  in  the  proportion  of  the  odd  num- 
bers 1,  3,  5,  &/c.,  for  the  lower  pulley,  and  corresponding 
grooves  on  the  face  of  another  solid  wheel  represented  by  the 
even  numbers  2,  4,  6,  &c.,  for  the  upper  pulley.  The  rope, 
being  passed  successively  over  the  grooves  of  such  wheels, 
would  be  thrown  off  exactly  in  the  same  manner  as  if  every 
groove  were  upon  a  separate  wheel,  and  every  wheel  revolved 
independently  of  the  others.  Such  is  White's  pulley,  repre- 
sented in  Jig.  121. 

The  advantage  of  this  machine,  when  accurately  construct- 
ed, is  very  considerable.     The  friction,  even  when  great  re- 


-  THE    ELEMENTS    OP    MECHANICS.  CHAP.  XV. 

sistances  are  to  be  opposed,  is  very  trifling ;  but,  on  the  other 
hand,  it,  has  corresponding  disadvantages  which  greatly  cir- 
cumscribe its  practical  utility.  In  the  workmanship  of  the 
grooves,  great  difficulty  is  found  in  giving  them  the  exact  pro- 
portions ;  in  doing  which,  the  thickness  of  the  rope  must  be 
accurately  allowed  for ;  and  consequently  it  follows,  that  the 
same  pulley  can  never  act,  except  with  a  rope  of  a  particular 
diameter.  A  very  slight  deviation  from  the  true  proportion 
of  the  grooves  will  cause  the  rope  to  be  unequally  stretched, 
and  will  throw  on  some  parts  of  it  an  undue  proportion  of 
the  weight,  while  other  parts  become  nearly,  and  sometimes 
altogether  slack.  Besides  these  defects,  the  rope  is  so  liable 
to  derangement  by  being  thrown  out  of  the  grooves,  that  the 
pulley  can  scarcely  be  considered  portable. 

For  these  and  other  reasons,  this  machine,  ingenious  as  it 
unquestionably  is,  has  never  been  extensively  used. 

(275.)  In  the  several  systems  of  pulleys  just  explained,  the 
hook  to  which  the  fixed  block  is  attached  supports  the  entire 
of  both  the  power  and  weight.  When  the  machine  is  in 
equilibrium,  the  power  only  supports  so  much  of  the  weight 
as  is  equal  to  the  tension  of  the  cord,  all  the  remainder  of  the 
weight  being  thrown  on  the  fixed  point,  according  to  what 
was  observed  in  (225.) 

If  the  power  be  moved  so  as  to  raise  the  weight,  it  will 
move  with  a  velocity  as  many  times  greater  than  that  of  the 
weight,  as  the  weight  itself  is  greater  than  the  power.  Thus 
in  jig.  118.,  if  the  weight  attached  to  the  lower  block  ascend 
one  foot,  six  feet  of  line  will  pass  over  the  pulley  A,  accord- 
ing to  what  has  been  already  proved.  Thus  the  power  will 
descend  through  six  feet,  while  the  weight  rises  one  foot. 
But,  in  this  case,  the  weight  is  six  times  the  power.  All  the 
observations  in  (226.)  will. therefore  be  applicable  to  the  cases 
of  great  weights  raised  by  small  powers  by  means  of  the  sys 
tern  of  pulleys  just  described. 

(276.)  When  two  or  more  ropes  are  used,  pulleys  may  be 
combined  in  various  ways  so  as  to  produce  any  degree  of 
mechanical  effect.  If  to  any  of  the  systems  already  describ- 
ed, a  single  movable  pulley  be  added,  the  power  of  the  ma- 
chine would  be  doubled.  In  this  case,  the  second  rope  is  at- 
tached to  the  hook  of  the  lower  block,  as  in  Jig.  122.,  and, 
being  carried  through  a  movable  pulley  attached  to  the 
weight,  it  is  finally  brought  up  to  a  fixed  point.  The  tension  of 
the  second  cord  is  equal  to  half  the  weight  (270.)  ;  and  there- 


CHAP.    XV.  SPANISH    BARTONS.  173 

fore  the  power  P,  by  means  of  the  first  cord,  will  have  only 
half  the  tension  which  it  would  have  if  the  weight  were  at- 
tached to  the  lower  block.  A  movable  pulley  thus  applied 
is  called  a  runner. 

(277.)  Two  systems  of  pulleys,  called  Spanish  bartons, 
having  each  two  ropes,  are  represented  in  Jig.  123.  The 
tension  of  the  rope  P  A  B  C  in  the  first  system  is  equal  to 
the  power ;  and  therefore  the  parts  B  A  and  B  C  support  a 
portion  of  the  weight  equal  to  twice  the  power.  The  rope 
E  A  supports  the  tensions  of  A  P  and  A  B ;  and  therefore 
the  tension  of  A  E  D  is  twice  the  power.  Thus  the  united 
tensions  of  the  ropes  which  support  the  pulley  B  is  four  times 
the  power,  which  is  therefore  the  amount  of  the  weight.  In 
the  second  system,  the  rope  P  A  D  is  stretched  by  the  power. 
The  rope  A  E  B  C  acts  against  the  united  tensions  A  P  and 
A  D ;  and  therefore  the  tension  of  A  E  or  E  B  is  twice  the 
power.  Thus  the  weight  acts  against  three  tensions ;  two  of 
which  are  equal  to  twice  the  power,  and  the  remaining  one  is 
equal  to  the  power.  The  weight  is  therefore  equal  to  five 
times  the  power. 

A  single  rope  may  be  so  arranged  with  one  movable  pul- 
ley as  to  support  a  weight  equal  to  three  times  the  power. 
In  Jig.  124.  this  arrangement  is  represented,  where  the  num- 
bers sufficiently  indicate  the  tension  of  the  rope,  and  the 
proportion  of  the  weight  and  power.  In  Jig.  125.  another 
method  of  producing  the  same  effect  with  two  ropes  is  repre- 
sented. 

(278.)  If  several  single  movable  pulleys  be  made  succes- 
sively to  act  upon  nach  other,  the  effect  is  doubled  by  every 
additional  pulley  :  such  a  system  as  this  is  represented  in  Jig. 
126.  The  tension  of  the  first  rope  is  equal  to  the  power  ;  the 
second  rope  acts  against  twice  the  tension  of  the  first,  and 
therefore  it  is  stretched  with  a  force  equal  to  twice  the  pow- 
er :  the  third  rope  acts  against  twice  this  tension,  and  there- 
fore it  is  stretched  with  a  force  equal  to  four  times  the  pow- 
er; and  so  on.  In  the  system  represented  in  Jig.  126.,  there 
are  three  ropes,  arid  the  weight  is  eight  times  the  power. 
Another  rope  would  render  it  sixteen  times  the  power,  and 
so  on. 

In  this  system,  it  is  obvious  that  the  ropes  will  require  to 
have  different  degrees  of  strength,  since  the  tension  to  which 
they  are  subject  increases  in  a  double  proportion   from  the 
power  to  the  weight. 
15  * 


174  THE  ELEMENTS  op  MECHANICS.  CHAP.  xv. 

(279.)  If  each  of  the  ropes,  instead  of  being  attached  to 
fixed  points  at  the  top,  are  carried  over  fixed  pulleys,  arid  at- 
tached to  the  several  movable  pulleys  respectively,  is  in  Jig. 
127.,  the  power  of  the  machine  will  be  greatly  increased  ;  for 
in  that  case  the  forces  which  stretch  the  successive  ropes  in- 
crease in  a  treble  instead  of  a  double  proportion,  as  will  be 
evident  by  attending  to  the  numbers  which  express  the  ten- 
sions in  the  figure.  One  rope  would  render  the  weight  three 
times  the  power ;  two  ropes  nine  limes;  three  ropes  twenty- 
seven  times;  and  so  on.  An  arrangement  of  pulleys  is  rep- 
resented in  Jig.  128.,  by  which  each  rope,  instead  of  being 
finally  attached  to  a  fixed  point,  as  hi  fig.  126.,  is  attached  to 
the  weight.  The  weight  is  in  this  case  supported  by  three 
ropes ;  one  stretched  with  a  force  equal  to  the  power ;  anoth- 
er with  a  force  equal  to  twice  the  power ;  and  a  third  with  a 
force  equal  to  four  times  the  power.  The  weight  is,  therefore, 
in  this  case,  seven  times  the  power. 

(280.)  If  the  ropes,  instead  of  being  attached  to  the  weight, 
pass  through  wheels,  as  in  Ji,g.  129.,  and  are  finally  attached 
to  the  pulleys  above,  the  power  of  the  machine  will  be  con- 
siderably increased.  In  the  system  here  represented,  the 
weight  is  twenty-six  times  the  power. 

(281.)  In  considering  these  several  combinations  of  pulleys, 
we  have  ^omitted  to  estimate  the  effects  produced  by  the 
weights  of  the  sheaves  and  blocks.  Without  entering  into 
the  details  of  this  computation,  it  may  be  observed  generally, 
that  in  the  systems  represented  in  fig*.  12(3.,  127.,  the  weight 
of  the  wheel  and  blocks  acts  against  the  power ;  but  that 
in  Jigs.  128.  and  129.  they  assist  the  powers  in  supporting 
the  weight.  In  the  systems  represented  in  Jig.  123.,  the 
weights  of  the  pulleys,  to  a  certain  extent,  neutralize  each 
other. 

(282.)  It  will  in  all  cases  be  found,  that  that  quantity  by 
which  the  weight  exceeds  the  power  is  supported  by  fixed 
points  ;  and  therefore,  although  it  be  commonly  stated  that  a 
small  power  supports  a  great  weight,  yet  in  the  pulley,  as  in 
all  other  machines,  the  power  supports  no  more  of  the  weight 
than  is  exactly  equal  to  its  own  amount.  It  will  not  be 
necessary  to  establish  this  in  each  of  the  examples  which 
have  been  given ;  having  explained  it  in  one  instance,  the 
student  will  find  no  difficulty  in  applying  the  same  reasoning 
to  others.  In  Jig.  126.,  the  fixed  pulley  sustains  a  force  equal 
to  twice  the  power,  and  by  it  the  oower  giving  tension  to  the 


CHAP.  XV.  PULLEYS.  175 

first  rope  sustains  a  part  of  the  weight  equal  to  itself.  The 
first  hook  sustains  a  portion  of  the  weight  equal  to  the  tension 
of  the  first  string,  or  to  the  power.  The  second  hook  sus- 
tains a  force  equal  to  twice  the  power ;  and  the  third  hook 
sustains  a  force  equal  to  four  times  the  power.  The  three 
hooks  therefore  sustain  a  portion  of  the  weight  equal  to  seven 
times  the  power ;  and  the  weight  itself  being  eight  times  the 
power,  it  is  evident  that  the  part  of  the  weight  which  re- 
mains to  be  supported  by  the  power,  is  equal  to  the  power 
itself. 

(283.)  When  a  weight  is  raised  by  any  of  the  systems  of 
pulleys  which  have  been  last  described,  the  proportion  between 
the  velocity  of  the  weight  and  the  velocity  of  the  power,  so 
frequently  noticed  in  other  machines,  will  always  be  observ- 
ed. In  the  system  of  pulleys  represented  in  jig.  126.,  the 
weight  being  eight  times  the  power,  the  velocity  of  the  pow- 
er will  be  eight  times  that  of  the  weight.  If  the  power  be  mov- 
ed through  eight  feet,  that  part  of  the  rop«  between  the  fixed 
pulley  and  the  first  movable  pulley  will  be  shortened  by 
eight  feet.  And  since  the  two  parts  which  lie  above  the  first 
movable  pulley  must  be  equally  shortened,  each  will  be  di- 
minished by  four  feet ;  therefore  the  first  pulley  will  rise 
through  four  feet,  while  the  power  moves  through  eight  feet. 
In  the  same  way  it  may  be  shown,  that  while  the  first  pulley 
moves  through  four  feet,  the  second  moves  through  two  ;  and 
while  the  second  moves  through  two,  the  third,  to  which  the 
weight  is  attached,  is  raised  through  one  foot.  While  the 
power,  therefore,  is  carried  through  eight  feet,  the  weight  is 
moved  through  one  foot. 

By  reasoning  similar  to  this,  it  may  be  shown  that  the  space 
through  which  the  power  is  moved  in  every  case  is  as  many 
times  greater  than  the  height  through  which  the  weight  is 
raised,  as  the  weight  is  greater  than  the  power. 

(284.)  From  its  portable  form,  cheapness  of  construetion, 
and  the  facility  with  which  it  may  be  applied  in  almost  every 
situation,  the  pulley  is  one  of  the  most  useful  of  the  simple 
machines.  The  mechanical  advantage,  however,  which  it 
appears  in  theory  to  possess,  is  considerably  diminished  in 
practice,  owing  to  the  stiffness  of  the  cordage,  and  the  friction 
of  the  wheels  and  blocks.  By  this  means  it  is  computed  that 
in  most  cases  so  great  a  proportion  as  two  thirds  of  the  power 
is  lost.  The  pulley  is  much  used  in  building,  where  weights 


176  THE    ELEMENTS    OF    MECHANICS.  CHAP.  XVI. 

are  to  be  elevated  to  great  heights.  But  its  most  extensive 
application  is  found  in  the  rigging  of  ships,  where  almost  eve- 
ry motion  is  accomplished  by  its  means. 

(285.)  In  all  the  examples  of  pulleys,  we  have  supposed 
the  parts  of  the  rope  sustaining  the  weight,  and  each  of  the 
movable  pulleys,  to  be  parallel  to  each  other.  If  they  be 
subject  to  considerable  obliquity,  the  relative  tensions  of  the 
different  ropes  must  be  estimated  according  to  the  principle 
applied  in  (271.) 


CHAPTER  XVI. 

ON    THE    INCLINED    PLANE,  WEDGE,  AND    SCREW. 

(286.)  THE  inclined  plane  is  the  most  simple  of  all  ma- 
chines. It  is  a  hard  plane  surface  forming  some  angle  with 
a  horizontal  plane,  that  angle  not  being  a  right  angle.  When 
a  weight  is  placed  on  such  a  plane,  a  two-fold  effect  is  pro- 
duced. A  part  of  the  effect  of  the  weight  is  resisted  by  the 
plane,  and  produces  a  pressure  upon  it ;  and  the  remainder 
urges  the  weight  down  the  plane,  and  would  produce  a  pres- 
sure against  any  .surface  resisting  its  motion  placed  in  a  direc- 
tion perpendicular  to  the  plane  (131.) 

Let  A  B,  .jig.  130.,  be  such  a  plane,  B  C  its  horizontal 
base,  A  C  its  height,  and  A  B  C  its  angle  of  elevation.  Let 
W  be  a  weight  placed  upon  it.  This  weight  acts  in  the  ver- 
tical direction  W  D,  and  is  equivalent  to  two  forces,  W  F 
perpendicular  to  the  plane,  and  W  E  directed  down  the  plane 
(74.)  If  a  plane  be  placed  at  right  angles  to  the  inclined 
plane  below  W,  it  will  resist  the  descent  of  »'ae  weight,  and 
sustain  a  pressure  expressed  by  W  E.  Thus  the  weight  W, 
resting  in  the  corner,  instead  of  producing  one  pressure  in  the 
direction  W  D,  will  produce  two  pressures,  one  expressed  by 
W  F  upon  the  inclined  plane,  and  the  other  expressed  by  W 
E  upon  the  resisting  plane.  These  pressures  respectively 
have  the  same  proportion  to  the  entire  weight,  as  W  F  and 
W  E  have  to  W  D,  or  as  D  E  and  W  E  have  to  W  D,  be- 
cause D  E  is  equal  to  W  F.  Now,  the  triangle  WED  is 
in  all  respects  similar  to  the  triangle  ABC,  the  one  differ- 
ing from  the  other  only  in  the  scale  on  which  it  is  construct- 


CHAP.  XVI.  INCLINED  ROADS.  177 

ed.  Therefore  the  three  lines  A  C,  C  B,  and  B  A,  are  in 
the  same  proportion  to  each  other  as  the  lines  W  E,  E  D,  and 
W  D.  Hence  A  B  has  to  A  C  the  same  proportion  as  the 
whole  weight  has  to  the  pressure  directed  towards  B,  and 
A  B  has  to  B  C  the  same  proportion  as  the  whole  weight  has 
to  the  pressure  on  the  inclined  plane. 

We  have  here  supposed  the  weight  to  be  sustained  upon 
the  inclined  plane,  by  a  hard  plane  fixed  at  right  angles  to  it. 
But  the  power  necessary  to  sustain  the  weight  will  be  the 
same,  in  whatever  way  it  is  applied,  provided  it  act  in  the  di- 
rection of  the  plane.  Thus  a  cord  may  be  attached  to  the 
weight,  and  stretched  towards  A,  or  the  hands  of  men  may 
be  applied  to  the  weight  below  it,  so  as  to  resist  its  descent 
towards  B.  But  in  whatever  way  it  be  applied,  the  amount  of 
the  power  will  be  determined  in  the  same  manner.  Suppose 
the  weight  to  consist  of  as  many  pounds  as  there  are  inches 
in  A  B,  then  the  power  requisite  to  sustain  it  upon  the  plane 
will  consist  of  as  many  pounds  as  there  are  inches  in  A  C, 
and  the  pressure  on  the  plane  will  amount  to  as  many  pounds 
as  there  are  inches  in  B  C. 

From  what  has  been  stated,  it  may  easily  be  inferred  that 
the  less  the  elevation  of  the  plane  is,  the  less  will  be  the  pow- 
er requisite  to  sustain  a  given  weight  upon  it,  and  the  greater 
will  be  the  pressure  upon  it.  Suppose  the  inclined  plane 
A  B  to  turn  upon  a  hinge  at  B,  and  to  be  depressed  so  that 
its  angle  of  elevation  shall  be  diminished,  it  is  evident  that  as 
this  angle  decreases,  the  height  of  the  plane  decreases,  and 
its  base  increases.  Thus,  when  it  takes  the  position  B  A', 
the  height  A'  C'  is  less  than  the  former  height  A  C,  while  the 
base  B  C'  is  greater  than  the  former  base  B  C.  The  power 
requisite  to  support  the  weight  upon  the  plane  in  the  position 
B  A'  is  represented  by  A'  C',  and  is  as  much  less  than  the 
power  requisite  to  sustain  it  upon  the  plane  A  B,  as  the  height 
A'  C'  is  less  than  the  height  A  C.  On  the  other  hand,  the 
pressure  upon  the  plane  in  the  position  B  A7  is  as  much  great- 
er than  the  pressure  upon  the  plane  B  A,  as  the  base  B  C'  is 
greater  than  the  base  B  C. 

(287.)  The  power  of  an  inclined  plane,  considered  as  a 
machine,  is  therefore  estimated  by  the  proportion  which  the 
length  bears  to  the  height.  This  power  is  always  increased 
by  diminishing  the  elevation  of  the  plane. 

Roads  which  are  not  lerel  may  be  regarded  as  inclined 
planes,  and  loads  drawn  upon  them  in  carriages,  considered 


178  THE    ELEMENTS    OF   MECHANICS.  CHAT.  XVI. 

in  reference  to  the  powers  which  impel  them,  are  subject  to 
all  the  conditions  which  have  been  established  for  inclined 
planes.  The  inclination  of  the  road  is  estimated  by  the 
height  corresponding  to  some  proposed  length.  Thus  it  is 
said  to  rise  one  foot  in  fifteen,  one  foot  in  twenty,  &,c.,  mean- 
ing that  if  fifteen  or  twenty  feet  of  the  road  be  taken  as  the 
length  of  an  inclined  plane,  such  as  A  B,  the  corresponding 
height  will  be  one  foot.  Or  the  same  may  be  expressed  thus  : 
that  if  fifteen  or  twenty  feet  be  measured  upon  the  roacj,  the 
difference  of  the  levels  of  the  two  extremities  of  the  distance 
measured  is  one  foot.  According  to  this  method  of  estimat- 
ing the  inclination  of  roads,  the  power  requisite  to  sustain  a 
load  upon  them  (setting  aside  the  effect  of  friction)  is  always 
proportional  to  that  elevation.  Thus,  if  a  road  rise  one  foot 
in  twenty,  a  power  of  one  ton  will  be  sufficient  to  sustain 
twenty  tons,  and  so  on. 

On  a  horizontal  plane,  the  only  resistance  which  the  power 
has  to  overcome,  is  the  friction  of  the  load  with  the  plane,  and 
the  consideration  of  this  being  for  the  present  omitted,  a 
weight  once  put  in  motion  would  continue  moving  for  ever, 
without  any  further  action  of  the  power.  But  if  the  plane  be 
inclined,  the  power  will  be  expended  in  raising  the  weight 
through  the  perpendicular  height  of  the  plane.  Thus,  in  a 
road  which  rises  one  foot  in  ten,  the  power  is  expended  in 
raising  the  weight  through  one  perpendicular  foot  for  every 
ten  feet  of  the  road  over  which  it  is  moved.  As  the  expendi- 
ture of  power  depends  upon  the  rate  at  which  the  weight  is 
raised  perpendicularly,  it  is  evident  that  the  greater  the  incli- 
nation of  the  road  is,  the  slower  the  motion  must  be  with  the 
same  force.  If  the  energy  of  the  power  be  such  as  to  raise 
the  weight  at  the  rate  of  one  foot  per  minute,  the  weight  may 
be  moved  in  each  minute  through  that  length  of  the  road 
which  corresponds  to  a  rise  of  one  foot.  Thus,  if  two  roads 
rise,  one  at  the  rate  of  a  foot  in  fifteen  feet,  and  the  other  at 
the  rate  of  one  foot  in  twenty  feet,  the  same  expenditure  of 
power  will  move  the  weight  through  fifteen  feet  of  the  one, 
and  twenty  feet  of  the  other  at  the  same  rate. 

From  such  considerations  as  these,  it  will  readily  appear 
that  it  may  often  be  more  expedient  to  carry  a  road  through  a 
circuitous  route  than  to  continue  it  in  the  most  direct  course ; 
for,  though  the  measured  length  of  road  may  be  considerably 
greater  in  the  former  case,  yet  more  may  be  gained  in 
speed  with  the  same  expenditure  of  power,  than  is  lost  by  the 


CKAP.    XVI.  INCLINED    PLANE.  179 

increase  of  distance.  By  attending  to  these  circumstances, 
modern  road-makers  have  greatly  facilitated  and  expedited 
the  intercourse  between  distant  places. 

(288.)  If  the  power  act  oblique  to  the  plane,  it  will  have  a 
twofold  effect ;  a  part  being  expended  in  supporting  or  draw- 
ing the  weight,  and  a  part  in  diminishing  or  increasing  the 
pressure  upon  the  plane.  Let  W  P,Jig.  130.,  be  the  power. 
This  will  be  equivalent  to  two  forces,  W  F',  perpendicular  to 
the  plane,  and  W  E'  in  the  direction  of  the  plane.  (74.)  In 
order  that  the  power  should  sustain  the  weight,  it  is  necessary 
that  that  part  W  E'  of  the  power  which  acts  in  the  direction 
of  the  plane,  should  be  equal  to  that  part  W  E,fg.  130.,  of 
the  weight  which  acts  down  the  plane.  The  other  part  W  F, 
of  the  power  acting  perpendicular  to  the  plane,  is  immediately 
opposed  to  that  part  W  F  of  the  weight  which  produces  pres- 
sure. The  pressure  upon  the  plane  will  therefore  be  dimin- 
ished by  the  amount  of  W  F'.  The  amount  of  the  power, 
which  will  equilibrate  with  the  weight,  may,  in  this  case,  be 
found  as  follows.  Take  W  E'  equal  to  W  E,  and  draw  E'  P 
perpendicular  to  the  plane,  and  meeting  the  direction  of  the 
power.  The  proportion  of  the  power  to  the  weight  will  be 
that  of  W  P  to  W  D.  And  the  proportion  of  the  pressure  to 
the  weight  will  be  that  of  the  difference  between  W  F  and 
W  F7  to  W  D.  If  the  amount  of  the  power  have  a  less  pro- 
portion to  the  weight  than  W  P  has  to  W  D,  it  will  not  sup- 
port the  body  on  the  plane,  but  will  allow  it  to  descend.  And 
if  it  had  a  greater  proportion,  it  will  draw  the  weight  up  the 
plane  towards  A. 

(289.)  It  somet'.nes  happens  that  a  weight  upon  one  in- 
clined plane  is  raised  or  supported  by  another  weight  upon 
another  inclined  plane.  Thus,  if  A  B  and  A  R',jig.  131.,  be 
two  inclined  planes,  forming  an  angle  at  A,  and  W  W  be 
two  weights  placed  upon  these  planes,  and  connected  by  a 
cord  passing  over  a  pulley  at  A,  the  one  weight  will  either 
sustain  the  other,  or  one  will  descend,  drawing  the  other  up. 
To  determine  the  circumstances  under  which  these  effects 
will  ensue,  draw  the  lines  W  D  and  W  D'  in  the  vertical  di- 
rection, and  take  upon  them  as  many  inches  as  there  are 
ounces  in  the  weights  respectively.  W  D  and  W  D'  being 
the  lengths  thus  taken,  and  therefore  representing  the  weights, 
the  lines  W  E  and  W  E'  will  represent  the  effects  of  these 
weights  respectively  down  the  planes.  If  WE  and  W  E'  be 
equal,  the  weights  will  sustain  each  other  without  motion. 


180  THE    ELEMENTS    OF   MECHANICS.  CHAP.    XVI 

But  if  W  E  be  greater  than  W  E',  the  weight  W  will  de- 
scend, drawing  the  weight  W'  up.  And  if  W  E7  be  greater 
than  W  E,  the  weight  W'  will  descend,  drawing  the  weight 
W  up.  In  every  case,  the  lines  W  F  and  W  F'  will  repre- 
sent the  pressures  upon  the  planes  respectively. 

It  is  not  necessary,  for  the  effect  just  described,  that  the 
inclined  planes  should,  as  represented  in  the  figure,  form  an 
angle  with  each  other.  They  may  be  parallel,  or  in  any  other 
position,  the  rope  being  carried  over  a  sufficient  number  of 
wheels  placed  so  as  to  give  it  the  necessary  deflection.  This 
method  of  moving  loads  is  frequently  applied  in  great  public 
works  where  rail-roads  are  used.  Loaded  wagons  descend 
one  inclined  plane,  while  other  wagons,  either  empty  or  so 
loaded  as  to  permit  the  descent  of  those  with  which  they  are 
connected,  are  drawn  up  the  other. 

(290.)  In  the  application  of  the  inclined  plane,  which  we 
have  hitherto  noticed,  the  machine  itself  is  supposed  to  be 
fixed  in  its  position,  while  the  weight  or  load  is  moved  upon 
it.  But  it  frequently  happens  that  resistances  are  to  be  over- 
come which  do  not  admit  to  be  thus  moved.  In  such  cases, 
instead  of  moving  the  load  upon  the  planes,  the  plane  is  to  be 
moved  under  or  against  the  load.  Let  D  E,^£g\  132.,  be  a 
heavy  beam  secured  in  a  vertical  position  between  guides,  F 
G  and  H  I,  so  that  it  is  free  to  move  upwards  and  downwards, 
but  not  laterally.  Let  A  B  C  be  an  inclined  plane,  the  ex- 
tremity of  which  is  placed  beneath  the  end  of  the  beam.  A 
force  applied  to  the  back  of  this  plane  A  C,  in  the  direction 
C  B,  will  .urge  the  plane  under  the  beam,  so  as  to  raise  the 
beam  to  the  position  represented  in  the^z°v  133.  Thus,  while 
the  inclined  plane  is  moved  through  the  distance  C  B,  the 
beam  is  raised  through  the  height  C  A. 

(291.)  When  the  inclined  plane  is  applied  in  this  manner, 
it  is  called  a  wedge.  And  if  the  power  applied  to  the  back 
were  a  continued  pressure,  its  proportion  to  the  weight  would 
be  that  of  A  C  to  C  B.  It  follows,  therefore,  that  the  more 
acute  the  angle  B  is,  the  more  powerful  will  be  the  wedge. 

In  some  cases,  the  wedge  is  formed  of  two  inclined  planes, 
placed  base  to  base,  as  represented  in  Jig.  134.  The  theoret- 
ical estimation  of  the  power  of  this  machine  is  not  applicable 
in  practice  with  any  degree  of  accuracy.  This  is  in  part 
owing  to  the  enormous  proportion  which  the  friction  in  most 
cases  bears  to  the  theoretical  value  of  the  power,  but  still 
more  to  the  nature  of  the  power  generally  used.  The  force 


CHAP;  xvi.  USES  OF  THE  WEDGE.  181 

of  a  blow  is  of  a  nature  so  wholly  different  from  continued 
forces,  such  as  the  pressure  of  weights,  or  the  resistance  of- 
fered by  the  cohesion  of  bodies,  that  they  admit  of  no  numer- 
ical comaprison.  Hence  we  cannot  properly  state  the  pro- 
portion which  the  force  of  a  blow  bears  to  the  amount  of  a 
weight  or  resistance.  The  wedge  is  almost  invariably  urged 
by  percussion ;  while  the  resistances  which  it  has  to  over- 
come are  as  constantly  forces  of  the  other  kind.  Although, 
however,  no  exact  numerical  comparison  can  be  made,  yet  it 
may  be  stated  in  a  general  way  that  the  wedge  is  more  and 
more  powerful  as  its  angle  is  more  acute. 

In  the  arts  and  manufactures,  wedges  are  used  where  enor- 
mous force  is  to  be  exerted  through  a  very  small  space. 
Thus  it  is  resorted  to  for  splitting  masses  of  timber  or  stone. 
Ships  are  raised  in  docks  by  wedges  driven  under  their  keels. 
The  wedge  is  the  principal  agent  in  the  oil-mill.  The  seeds 
from  which  the  oil  is  to  be  extracted  are  introduced  into  hair 
bags,  and  placed  between  planes  of  hard  wood.  Wedges  in- 
serted between  the  bags  are  driven  by  allowing  heavy  beams 
to  fall  on  them.  The  pressure  thus  excited  is  so  intense,  that 
the  seeds  in  the  bags  are  formed  into  a  mass  nearly  as  solid 
as  wood. 

Instances  have  occurred  in  which  the  wedge  has  been  used 
to  restore  a  tottering  edifice  to  its  perpendicular  position.  All 
cutting  and  piercing  instruments,  such  as  knives,  razors,  scis- 
sors, chisels,  &LC.,  nails,  pins,  needles,  awls,  &,c.,  are  wedges. 
The  angle  of  the  wedge,  in  these  cases,  is  more  or  less  acute, 
according  to  the  purpose  to  which  it  is  to  be  applied.  In  de- 
termining this,  two  things  are  to  be  considered — the  mechan- 
ical power,  which  is  increased  by  diminishing  the  angle  of  the 
wedge,  and  the  strength  of  the  tool,  which  is  always  dimin- 
ished by  the  same  cause.  There  is,  therefore,  a  practical 
limit  to  the  increase  of  the  power,  and  that  degree  of  sharp- 
ness only  is  to  be  given  to  the  tool  which  is  consistent  with 
the  strength  requisite  for  the  purpose  to  which  it  is  to  be  ap- 
plied. In  tools  intended  for  cutting  wood,  the  angle  is  gen- 
erally about  30°.  For  iron  it  is  from  50°  to  60° ;  and  for 
brass,  from  80°  to  90°.  Tools  which  act  by  pressure  may  be 
made  more  acute  than  those  which  are  driven  by  a  blow ; 
and,  in  general,  the  softer  and  more  yielding  the  substance  to 
be  divided  is,  and  the  less  the  power  required  to  act  upon  it, 
the  more  acute  the  wedge  may  be  constructed. 

In  many  cases,  the  utility  of  the  wedge  depends  on  that 
16 


18S  THE    ELEMENTS   OP    MECHANICS.  CHAP.    XVI. 

which  is  entirely  omitted  in  its  theory,  viz.  the  friction  which 
arises  between  its  surface  and  the  substance  which  it  divides. 
This  is  the  case  when  pins,  bolts  or  nails  are  used  for  bind- 
ing the  parts  of  structures  together  ;  in  which  case,  were  it 
not  for  the  friction,  they  would  recoil  from  their  places,  and 
fail  to  produce  the  desired  effect.  Even  when  the  wedge  is 
used  as  a  mechanical  engine,  the  presence  of  friction  is  ab- 
solutely indispensable  to  its  practical  utility.  The  power,  as 
has  already  been  stated,  generally  acts  by  successive  blows, 
and  is  therefore  subject  to  constant  intermission,  and,  but  for 
the  friction,  the  wedge  would  recoil  between  the  intervals  of 
the  blows  with  as  much  force  as  it  had  been  driven  forward. 
Thus  the  object  of  the  labor  would  be  continually  frustrated. 
The  friction,  in  this  case,  is  of  the  same  use  as  a  ratchet 
wheel,  but  is  much  more  necessary,  as  the  power  applied  to 
the  wedge  is  more  liable  to  intermission  than  in  the  cases 
where  ratchet  wheels  are  generally  used. 

(292.)  When  a  road  directly  ascends  the  side  of  a  hill,  it 
is  to  be  considered  as  an  inclined  plane ;  but  it  will  not  lose 
its  mechanical  character,  if,  instead  of  directly  ascending 
towards  the  top  of  the  hill,  it  winds  successively  round  it,  and 
gradually  ascends,  so  as,  after  several  revolutions,  to  reach  the 
top.  In  the  same  manner  a  path  may  be  conceived  to  sur- 
round a  pillar  by  which  the  ascent  may  be  facilitated  upon 
the  principle  of  the  inclined  plane.  Winding  stairs  con- 
structed in  the  interior  of  great  columns  partake  of  this 
character  ;  for  although  the  ascent  be  produced  by  successive 
steps,  yet  if  a  floor  could  be  made  sufficiently  rough  to  pre- 
vent the  feet  from  slipping,  the  ascent  would  be  accomplished 
with  equal  facility.  In  such  a  case,  the  winding  path  would 
be  equivalent  to  an  inclined  plane,  bent  into  such  a  form  as 
to  accommodate  it  to  the  peculiar  circumstances  in  which  it 
would  be  required  to  be  used.  It  will  not  be  difficult  to  trace 
the  resemblance  between  such  an  adaptation  of  the  inclined 
plans  and  the  appearances  presented  by  the  thread  of  a  screw: 
and  it  may  hence  be  easily  understood  that  a  screw  is  nothing 
more  than  an  inclined  plane  constructed  upon  the  surface  of 
a  cylinder. 

This  will,  perhaps,  be  more  apparent  by  the  following  con- 
trivance :  Let  A  B,/g\  135.,  be  a  common  round  ruler,  and 
let  C  D  E  be  a  piece  of  white  paper  cut  in  the  form  of  an 
inclined  plane,  whose  height  C  D  is  equal  to  the  length  of 
the  ruler  A  B,  and  let  the  edge  C  E  of  the  paper  be  marked 


CHAP.    XVI,  THE    SCREW,  "  183 

with  a  broad  black  line  :  let  the  edge  C  D  be  applied  to  the 
ruler  A  B}  and,  being  attached  thereto,  let  the  paper  be 
rolled  round  the  ruler ;  the  ruler  will  then  present  the  appear- 
ance of  a  screw,  ^g1.  136,,  the  thread  of  the  screw  being 
marked  by  the  black  line  C  E,  winding  continually  round 
the  ruler.  Let  D  F,  jig*  135.,  be  equal  to  the  circumference 
of  the  ruler,  and  draw  F  G  parallel  to  D  C,  and  G  H  parallel 
to  D  E,  the  part  C  G  F  D  of  the  paper  will  exactly  surround 
the  ruler  once ;  the  part  C  G  will  form  one  spire  of  the 
thread,  and  may  be  considered  as  the  length  of  one  inclined 
plane  surrounding  the  cylinder,  C  H  being  the  corresponding 
height,  and  G  H  the  base.  The  power  of  the  screw  does 
not,  as  in  the  ordinary  cases  of  the  inclined  plane,  act  par- 
allel to  the  plane  or  thread,  but  at  right1  angles  to  the  length 
of  the  cylinder  A  B,  or,  what  is  to  the  same  effect,  parallel 
to  the  base  H  G ;  therefore  the  proportion  of  the  power  to 
the  weight  will  be,  according  to  principles  already  explained, 
the  same  as  that  of  C  H  to  the  space  through  which  the 
power  moves  parallel  to  H  G  in  one  revolution  of  the  screw. 
H  C  is  evidently  the  distance  between  the  successive  positions 
of  the  thread  as  it  winds  round  the  cylinder  ;  and  it  appears, 
from  what  has  been  just  stated,  that  the  less  this  distance  is, 
or,  in  other  words,  the  finer  the  thread  is,  the  more  powerful 
the  machine  will  be. 

(293.)  In  the  application  of  the  screw,  the  weight  or  re- 
sistance is  not,  as  in  the  inclined  plane  and  wedge,  placed 
upon  the  surface  of  the  plane  or  thread.  The  power  is  usu- 
ally transmitted  by  causing  the  screw  to  move  in  a  concave 
cylinder,  on  the  interior  surface  of  which  a  spiral  cavity  is 
cut,  corresponding  exactly  to  the  thread  of  the  screw,  and 
in  which  the  thread  will  move  by  turning  round  the  screw 
continually  in  the  same  direction.  This  hollow  cylinder  is 
usually  called  the  nut  or  concave  screw.  The  screw  surrounded 
by  its  spiral  thread  is  represented  iiijig.  137. ;  and  a  section 
of  the  same  playing  in  the  nut  is  represented  in  Jig.  138. 

There  are  several  ways  in  which  the  effect  of  the  power 
may  be  conveyed  to  the  resistance  by  this  apparatus. 

First,  let  us  suppose  that  the  nut  A  B  is  fixed.  If  the 
screw  be  continually  turned  on  its  axis,  by  a  lever  E  F  in- 
serted in  one  end  of  it,  it  will  be  moved  in  the  direction 
C  D,  advancing  every  revolution  through  a  space  equal  to  the 
distance  between  two  contiguous  threads.  By  turning  the 
lever  in  an  opposite  direction,  the  screw  will  be  moved  in  the 
direction  D  C. 


184  THE    ELEMENTS    OF    MECHANICS.  CHAP.  XVI 

It  the  screw  be  fixed,  so  as  to  be  incapable  either  of  moving 
longitudinally  or  revolving  on  its  axis,  the  nut  A  B  may  be 
turned  upon  the  screw  by  a  lever,  and  will  move  on  the 
screw  towards  C  or  towards  D,  according  to  the  direction  in 
which  the  lever  is  turned. 

In  the  former  case,  we  have  supposed  the  nut  to  be  abso- 
lutely immovable,  and  in  the  latter  case,  the  screw  to  be 
absolutely  immovable.  It  may  happen,  however,  that  the 
nut,  though  capable  of  revolving,  is  incapable  of  moving  lon- 
gitudinally ;  and  that  the  screw,  though  incapable  of  revolving, 
is  capable  of  moving  longitudinally.  In  that  case,  by  turn- 
ing the  nut  A  B  upon  the  screw  by  the  lever,  the  screw  will 
be  urged  in  the  direction  C  D  or  DC,  according  to  the  way 
in  which  the  nut  is  turned. 

The  apparatus  may,  on  the  contrary,  be  so  arranged,  that 
the  nut,  though  incapable  of  revolving,  is  capable  of  moving 
longitudinally ;  and  the  screw,  though  capable  of  revolving, 
is  incapable  of  moving  longitudinally.  In  this  case,  by  turn- 
ing the  screw  in  the  one  direction  or  in  the  other,  the  nut 
A  B  will  be  urged  in  the  direction  C  D  or  D  C. 

All  these  various  arrangements  may  be  observed  in  differ- 
ent applications  to  the  machine. 

(294.)  A  screw  may  be  cut  upon  a  cylinder  by  placing 
the  cylinder  in  a  turning  lathe,  and  giving  it  a  rotatory  mo- 
tion upon  its  axis.  The  cutting  point  is  then  presented  to 
the  cylinder,  and  moved  in  the  direction  of  its  length,  at 
such  a  rate  as  to  be  carried  through  the  distance  between  the 
intended  thread,  while  the  cylinder  revolves  once.  The  rel- 
ative motions  of  the  cutting  point  and  the  cylinder  being 
preserved  with  perfect  uniformity,  the  thread  will  be  cut 
from  one  end  to  the  other.  The  shape  of  the  threads  may  be 
either  square,  as  in  Jig.  137.,  or  triangular,  as  in  Jig.  139. 

(295.)  The  screw  is  generally  used  in  cases  where  severe 
pressure  is  to  be  excited  through  small  spaces ;  it  is  therefore 
the  agent  in  most  presses,  in  Jig.  140.,  the  nut  is  fixed,  and 
by  turning  the  lever,  which  passes  through  the  head  of  the 
screw,  a  pressure  is  excited  upon  any  substance  placed  upon 
the  plate  immediately  under  the  end  of  the  screw.  In 
jig.  141.,  the  screw  is  incapable  of  revolving,  but  is  capable 
of  advancing  in  the  direction  of  its  length.  On  the  other 
hand,  the  nut  is  capable  of  revolving,  but  does  not  advance 
in  the  direction  of  the  screw.  When  the  nut  is  turned  by 


CHAP.    XVI.  USES    OF    THE    SCREW.  185 

means  of  the  screw  inserted  in  it,  the  screw  advances  in  the 
direction  of  its  length,  and  urges  the  board  which  is  attached 
to  it  upwards,  so  as  to  press  any  substance  placed  between 
it  and  the  fixed  board  above. 

In  cases  where  liquids  or  juices  are  to  be  expressed  from 
solid  bodies,  the  screw  is  the  agent  generally  employed.  It 
is  also  used  in  coining,  where  the  impression  of  a  dye  is  to 
be  made  upon  a  piece  of  metal,  and  in  the  same  way  in  pro- 
ducing the  impression  of  a  seal  upon  wax  or  other  substance 
adapted  to  receive  it.  When  soft  and  light  materials,  such 
as  cotton,  are  to  be  reduced  to  a  convenient  bulk  for  transpor- 
tation, the  screw  is  used  to  compress  them,  and  they  are  thus 
reduced  into  hard,  dense  masses.  In  printing,  the  paper  is 
urged  by  a  severe  and  sudden  pressure  upon  the  types,  by 
means  of  a  screw. 

(296.)  As  the  mechanical  power  of  the  screw  depends 
upon  the  relative  magnitude  of  the  circumference  through 
which  the  power  revolves,  and  the  distance  between  the 
threads,  it  is  evident,  that,  to  increase  the  efficacy  of  the 
machine,  we  must  either  increase  the  length  of  the  lever  by 
which  the  power  acts,  or  diminish  the  magnitude  of  the 
thread.  Although  there  is  no  limit  in  theory  to  the  increase 
of  the  mechanical  efficacy  by  these  means,  yet  practical  in- 
convenience arises  which  effectually  prevents  that  increase 
being  carried  beyond  a  certain  extent.  If  the  lever  by 
which  the  power  acts  be  increased,  the  same  difficulty  arises 
as  was  already  explained  in  the  wheel  and  axle  (254.) ;  the 
space  through  which  the  power  should  act  would  be  so  un- 
wieldy, that  its  application  would  become  impracticable.  If, 
on  the  other  hand,  the  power  of  the  machine  be  increased 
by  diminishing  the  size  of  the  thread,  the  strength  of  the 
thread  will  be  so  diminished,  that  a  slight  resistance  will 
tear  it  from  the  cylinder.  The  cases  in  which  it  is  necessary 
to  increase  the  power  of  the  machine  being  those  in  which 
the  greatest  resistances  are  to  be  overcome,  the  object  will 
evidently  be  defeated,  if  the  means  chosen  to  increase  that 
power  deprive  the  machine  of  the  strength  which  is  necessary 
to  sustain  the  force  to  which  it  is  to  be  submitted. 

(297.)  These  inconveniences  are  removed  by  a  contrivance 
of  Mr.  Hunter,  which,  while  it  gives  to  the  machine  all  the 
requisite  strength  and  compactness,  allows  it  to  have  an 
almost  unlimited  degree  of  mechanical  efficacy. 

This  contrivance  consists  in  the  use  of  two  screws,  the 
16* 


186  THE  ELEMENTS  OP  MECHANICS.     CHAP.  XVI. 

threads  of  which  may  have  any  strength  and  magnitude,  but 
which  have  a  very  small  difference  of  breadth.  While  the 
working  point  is  urged  forward  by  that  which  has  the  greater 
thread,  it  is  drawn  back  by  that  which  has  the  less ;  so  that, 
during  each  revolution  of  the  screw,  instead  of  being  ad- 
vanced through  a  space  equal  to  the  magnitude  of  either  of 
the  threads,  it  moves  through  a  space  equal  to  their  differ- 
ence. The  mechanical  power  of  such  a  machine  will  be  the 
same  as  that  of  a  single  screw  having  a  thread,  whose  mag- 
nitude is  equal  to  the  difference  of  the  magnitudes  of  the 
two  threads  just  mentioned. 

Thus,  without  inconveniently  increasing  the  sweep  of  the 
power,  on  the  one  hand,  or,  on  the  other,  diminishing  the 
thread  until  the  necessary  strength  is  losj;,  the  machine  will 
acquire  an  efficacy  limited  by  nothing  but  the  smallness  of 
the  difference  between  the  two  threads. 

This  principle  was  first  applied  in  the  manner  represented 
hi  Jig.  142.  A  is  the  greater  thread,  playing  in  the  fixed 
nut;  B  is  the  lesser  thread,  cut  upon  a  smaller  cylinder,  and 
playing  in  a  concave  screw,  cut  within  the  greater  cylinder. 
During  every  revolution  of  the  screw,  the  cylinder  A  de- 
scends through  a  space  equal  to  the  distance  between  its 
threads.  At  the  same  time  the  smaller  cylinder  B  ascends 
through  a  space  equal  to  the  distance  between  the  threads 
cut  upon  it :  the  effect  is,  that  the  board  D  descends  through 
a  space  equal  to  the  difference  between  the  threads  upon  A 
and  the  threads  upon  B,  and  the  machine  has  a  power  pro- 
portionate to  the  smallness  of  this  difference. 

Thus,  suppose  the  screw  A  has  twenty  threads  in  an  inch, 
while  the  screw  B  has  twenty-one;  during  one  revolution, 
the  screw  A  will  descend  through  a  space  equal  to  the  20th 
part  of  an  inch.  If,  during  this  motion,  the  screw  B  did  not 
turn  within  A,  the  board  D  would  be  advanced  through  the 
20th  of  an  inch ;  but  because  the  hollow  screw  within  A 
turns  upon  B,  the  screw  B  will,  relatively  to  A,  be  raised  in 
one  revolution  through  a  space  equal  to  the  21st  part  of  an 
inch.  Thus,  while  the  board  D  is  depressed  through  the  20th 
of  an  inch  by  the  screw  A,  it  is  raised  through  the  21st  of 
an  inch  by  the  screw  B.  It  is,  therefore,  on  the  whole,  de- 
pressed through  a  space  equal  to  the  excess  of  the  £0th  ot 
an  inch  above  the  21st  of  an  inch,  that  is,  through  the  420th 
of  an  inch. 

The  power  of  this  machine  will,  therefore,  be  expressed  by 


CHAP.  xvi.  HUNTER'S  SCREW.  187 

the  number  of  times  the  420th  of  an  inch  is  contained  in 
the  circumference  through  which  the  power  moves. 

(298.)  In  the  practical  application  of  this  principle  at 
present,  the  arrangement  is  somewhat  different.  The  two 
threads  are  usually  cut  on  different  parts  of  the  same  cylinder. 
If  nuts  be  supposed  to  be  placed  upon  these,  which  are  capa- 
ble of  moving  in  the  direction  of  the  length,  but  not  of  re- 
volving, it  is  evident  that  by  turning  the  screw  once  round, 
each  nut  will  be  advanced  through  a  space  equal  to  the 
breadth  of  the  respective  threads.  By  this  means  the  two 
nuts  will  either  approach  each  other,  or  mutually  recede,  ac- 
cording to  the  direction  in  which  the  screw  is  turned,  through 
a  space  equal  to  the  difference  of  the  breadth  of  the  threads, 
and  they  will  exert  a  force  either  in  compressing  or  extending 
any  substance  placed  between  them,  proportionate  to  the 
smallness  of  that  difference. 

(299.)  A  toothed  wheel  is  sometimes  used  instead  of  a  nut, 
so  that  the  same  quality  by  which  the  revolution  of  the 
screw  urges  the  nut  forward  is  applied  to  make  the  wheel 
revolve.  The  screw  is  in  this  case  called  an  endless  screw, 
because  its  action  upon  the  wheel  may  be  continued  without 
limit.  This  application  of  the  screw  is  represented  in 
Jig.  143.  P  is  the  winch  to  which  the  power  is  applied ; 
and  its  effect  at  the  circumference  of  the  wheel  is  estimated 
in  the  same  manner  as  the  effect  of  the  screw  upon  the  nut. 
This  effect  is  to  be  considered  as  a  power  acting  upon  the 
circumference  of  the  wheel ;  and  its  proportion  to  the  weight 
or  resistance  is  to  be  calculated  in  the  same  manner  as  the 
oroportion  of  the  power  to  the  weight  in  the  wheel  and  axle. 

300.  We  have  hitherto  considered  the  screw  as  an  engine 
used  to  overcome  great  resistances.  It  is  also  eminently 
useful  in  several  departments  of  experimental  science,  for 
the  measurement  of  very  minute  motions  and  spaces,  the 
magnitude  of  which  could  scarcely  be  ascertained  by  any 
other  means.  The  very  slow  motion  which  may  be  imparted 
to  the  end  of  a  screw,  by  a  very  considerable  motion  in  the 
power,  renders  it  peculiarly  well  adapted  for  this  purpose. 
To  explain  the  manner  in  which  it  is  applied — suppose  a 
screw  to  be  so  cut  as  to  have  fifty  threads  in  an  inch,  each 
revolution  of  the  screw  will  advance  its  point  through  the 
fiftieth  part  of  an  inch.  Now,  suppose  the  head  of  the 
screw  to  be  a  circle,  whose  diameter  is  an  inch,  the  circum- 
ference of  the  head  will  be  something  more  than  three  inches  : 


188  THE  .ELEMENTS  OF  MECHANICS.  CHAP.  XVI. 

this  may  be  easily  divided  into  a  hundred  equal  parts  distinct- 
ly visible.  If  a  fixed  index  be  presented  to  this  graduated 
circumference,  the  hundredth  part  of  a  revolution  of  the 
screw  may  be  observed,  by  noting  the  passage  of  one  division 
of  the  head  under  the  index.  Since  one  entire  revolution 
of  the  head  moves  the  point  through  the  fiftieth  of  an  inch, 
one  division  will  correspond  to  the  five  thousandth  of  an  inch. 
In  order  to  observe  the  motion  of  the  point  of  the  screw  in 
this  case,  a  fine  wire  is  attached  to  it,  which  is  carried  across 
the  field  of  view  of  a  powerful  microscope,  by  which  the  mo- 
tion is  so  magnified  as  to  be  distinctly  perceptible. 

A  screw  used  for  such  purposes  is  called  a  micrometer 
screw.  Such  an  apparatus  is  usually  attached  to  the  limbs  of 
graduated  instruments,  for  the  purposes  of  astronomical  and 
other  observation.  Without  the  aid  of  this  apparatus,  no 
observation  could  be  taken  with  greater  accuracy  than  the 
amount  of  the  smallest  div  ision  upon  the  limb.  Thus,  if  an 
instrument  for  measuring  angles  were  divided  into  small 
arches  of  one  minute,  and  an  angle  were  observed  which 
brought  the  index  of  the  instrument  to  some  point  between 
two  divisions,  we  could  only  conclude  that  the  observed 
angle  must  consist  of  a  certain  number  of  degrees  and 
minutes,  together  with  an  additional  number  of  seconds, 
which  would  be  unknown,  inasmuch  as  there  would  be  no 
means  of  ascertaining  the  fraction  of  a  minute  between  the 
index  and  the  adjacent  division  of  the  instrument.  But  if  a 
screw  be  provided,  the  point  of  which  moves  through  a  space 
equal  to  one  division  of  the  instrument,  with  sixty  revolu- 
tions of  the  head,  and  the  head  itself  be  divided  into  one 
hundred  equal  parts,  each  complete  revolution  of  the  screw 
will  correspond  to  the  sixtieth  part  of  a  minute,  or  to  one 
second,  and  each  division  on  the  head  of  the  screw  will  cor- 
respond to  the  hundredth  part  of  a  second.  The  index  being 
attached  to  this  screw,  let  the  head  be  turned  until  the  index 
be  moved  from  its  observed  position  to  the  adjacent  division 
of  the  limb.  The  number  of  complete  revolutions  of  the 
screw  necessary  to  accomplish  this  will  be  the  number  of 
seconds :  and  the  number  of  parts  of  a  revolution  over  the 
complete  number  of  revolutions  will  be  the  hundredth  parts 
of  a  second  necessary  to  be  added  to  the  degrees  and  minutes 
primarily  observed. 

It  is  not,  however,  only  to  angular  instruments  that  the 
micrometer  screw  is  applicable  ;  any  spaces  whatever  may  be 


CHAP.    XVII.  REGULATION  OF  MACHINES.  189 

measured  by  it.  An  instance  of  its  mechanical  application 
may  be  mentioned  in  a  steel-yard,  an  instrument  for  ascer- 
taining the  amount  of  weights  by  a  given  weight,  sliding  on 
a  long  graduated  arm  of  a  lever.  The  distance  from  the 
fulcrum,  at  which  this  weight  counterpoises  the  weight  to  be 
ascertained,  serves  as  a  measure  to  the  amount  of  that  weight. 
When  the  sliding  weight  happens  to  be  placed  between  two 
divisions  of  the  arm,  a  micrometer  screw  is  used  to  ascertain 
the  fraction  of  the  division. 

Hunter's  screw,  already  described,  seems  to  be  well  adapt- 
ed to  micrometrical  purposes  ;  since  the  motion  of  the  point 
may  be  rendered  indefinitely  slow,  without  requiring  an  ex- 
quisitely fine  thread,  such  as,  in  the  single  screw,  would  in 
this  case  be  necessary. 


CHAPTER  XVII. 

ON  THE  REGULATION  AND  ACCUMULATION  OF  FORCE. 

(301.)  IT  is  frequently  indispensable,  and  always  desirable, 
that  the  operation  of  a  machine  should  be  regular  and  uni- 
form. Sudden  changes  in  its  velocity,  and  desultory  varia- 
tions in  the  effective  energy  of  its  power,  are  often  injurious 
or  destructive  to  the  apparatus  itself,  and  when  applied  to 
manufactures,  never  fail  to  produce  unevenness  in  the  work. 
To  invent  methods  for  insuring  the  regular  motion  of  ma- 
chinery, by  removing  those  causes  of  inequality  which  may 
be  avoided,  and  by  compensating  others,  has  therefore  been 
a  problem  to  which  much  attention  and  ingenuity  have  been 
directed.  This  is  chiefly  accomplished  by  controlling,  and, 
as  it  were,  measuring  out,  the  power  according  to  the  exi- 
gencies of  the  machine,  and  causing  its  effective  energy  to 
be  always  commensurate  with  the  resistance  which  it  has  to 
overcome. 

Irregularity  in  the  motion  of  machinery  may  proceed  from 
one  or  more  of  the  following  causes  : — 1.  irregularity  in  the 
prime  mover ;  2.  occasional  variation  in  the  amount  of  the 
load  or  resistance  ;  and,  3.  because,  in  the  various  positions 
which  the  parts  of  the  machine  assume  during  its  motion, 
the  power  may  not  be  transmitted  with  equal  effect  to  the 
working  point. 


190  THE  ELEMENTS  OF  MECHANICS.  CHAP.  XVII. 

The  energy  of  the  prime  mover  is  seldom,  if  ever^  regular. 
The  force  of  water  varies  with  the  copiousness  of  the  stream. 
The  power  which  impels  the  windmill  is  proverbially  capri- 
cious. The  pressure  of  steam  varies  with  the  intensity  of 
the  furnace.  Animal  power,  the  result  of  will,  temper,  and 
health,  is  difficult  of  control.  -Human  labor  is  most  of  all 
unmanageable  ;  and  no  machine  works  so  irregularly  as  one 
which  is  manipulated.  In  some  cases,  the  moving  force  is 
subject,  by  the  very  conditions  of  its  existence,  to  constant 
variation,  as  in  the  example  of  a  spring,  which  gradually 
loses  its  energy  as  it  recoils.  (255).  In  many  instances,  the 
prime  mover  is  liable  to  regular  intermission,  and  is  actually 
suspended  for  certain  intervals  of  time.  This  is  the  case  in 
the  single  acting  steam-engine,  where  the  pressure  of  the 
steam  urges  the  descent  of  the  piston,  but  is  suspended  dur- 
ing its  ascent. 

The  load  or  resistance  to  which  the  machine  is  applied 
is  not  less  fluctuating.  In  mills,  there  are  a  multiplicity  of 
parts  which  are  severally  liable  to  be  occasionally  disengaged, 
and  to  have  their  operation  suspended.  In  large  factories 
for  spinning,  weaving,  printing,  &,c.,  a  great  number  of  sepa- 
rate spinning  machines,  looms,  presses,  or  other  engines,  are 
usually  worked  by  one  common  mover,  such  as  a  water-wheel 
or  steam-engine.  In  these  cases,  the  number  of  machines 
employed  from  time  to  time  necessarily  varies  with  the  fluc- 
tuating demand  for  the  articles  produced,  and  from  other 
causes.  Under  such  circumstances,  the  velocity  with  which 
every  part  of  the  machinery  is  moved  would  suffer  corre- 
sponding changes,  increasing  its  rapidity  with  every  augmenta- 
tion of  the  moving  power  or  diminution  of  the  resistance,  or 
being  retarded  in  its  speed  by  the  contrary  circumstances. 

But  even  when  the  prime  mover  and  the  resistance  are 
both  regular,  or  rendered  so  by  proper  contrivances,  still  it 
will  rarely  happen  that  the  machine  by  which  the  energy  of 
the  one  is  transmitted  to  the  other  conveys  this  with  unim- 
paired effect  in  all  the  phases  of  its  operation.  To  give  a 
general  notion  of  this  cause  of  inequality,  to  those  who  have 
not  been  familiar  with  machinery,  would  not  be  easy,  without 
having  recourse  to  an  example.  For  the  present  we  shall 
merely  state,  that  the  several  moving  parts  of  every  machine 
assume  in  succession  a  variety  of  positions  ;  that  at  regular 
periods  they  return  to  their  first  position,  and  again  undergo 
the  same  succession  of  changes.  In  the  different  positions 


CHAP.  XVII.  REGULATORS. GOVERNOR.  191 

through  which  they  are  carried  in  every  period  of  motion,  the 
efficacy  of  the  machine  to  transmit  the  power  to  the  resist- 
ance is  different,  and  thus  the  effective  energy  of  the  ma- 
chine in  acting  upon  the  resistance  would  be  subject  to  con- 
tinual fluctuation.  This  will  be  more  clearly  understood 
when  we  come  to  explain  the  methods  of  counteracting  the 
defect  or  equalizing  the  action  of  the  power  upon  the  re- 
sistance. 

Such  are  the  chief  causes  of  the  inequalities  incidental 
to  the  motion  of  machinery,  and  we  now  propose  to  describe 
a  few  of  the  many  ingenious  contrivances  which  the  skill 
of  engineers  has  produced  to  remove  the  consequent  incon- 
veniences. 

(302.)  Setting  aside,  for  the  present,  the  last  cause  of  ine- 
quality, and  considering  the  machinery,  whatever  it  be,  to 
transmit  the  power  to  the  resistance  without  irregular  inter- 
ruption, it  is  evident  that  every  contrivance,  having  for  its 
object  to  render  the  velocity  uniform,  can  only  accomplish 
this  by  causing  the  variations  of  the  power  and  resistance  to 
be  proportionate  to  each  other.  This  may  be  done  either 
by  increasing  or  diminishing  the  power  as  the  resistance 
increases  or  diminishes  ;  or  by  increasing  or  diminishing  the 
resistance  as  the  power  increases  or  diminishes  ;  and  accord- 
ing to  the  facilities  or  convenience  presented  by  the  pecu- 
liar circumstances  of  the  case,  either  of  these  methods  is 
adopted. 

The  contrivances  for  effecting  this  are  called  regulators, 
Most  regulators  act  upon  that  part  of  the  machine  which 
commands  the  supply  of  the  power  by  means  of  levers,  or 
some  other  mechanical  contrivance,  so  as  to  check  the  quan- 
tity of  the  moving  principle  conveyed  to  the  machine  when 
the  velocity  has  a  tendency  to  increase  j  and,  on  the  other 
hand,  to  increase  that  supply  upon  any  undue  abatement  of 
its  speed.  In  a  water-mill  this  is  done  by  acting  upon  the 
shuttle  ;  in  a  wind-mill,  by  an  adjustment  of  the  sail-cloth  ; 
and  in  a  steam-engine,  by  opening  or  closing,  in  a  greater  or 
less  degree,  the  valve  by  which  the  cylinder  is  supplied  with 
steam. 

(303.)  Of  all  the  contrivances  for  regulating  machinery, 
that  which  is  best  known  and  most  commonly  used  is  the 
governor.  This  regulator,  which  had  been  long  in  use  in 
mill-work  and  other  machinery,  has  of  late  years  attracted 
more  general  notice  by  its  beautiful  adaptation  in  the  steam- 


192  THE  ELEMENTS  OF  MECHANICS.  CHAP.  XVII. 

engines  of  Watt.  It  consists  of  heavy  balls  B  B,  Jig.  144., 
attached  to  the  extremities  of  rods  B  F.  These  rods  play 
upon  a  joint  at  E,  passing  through  a  mortise  in  the  vertical 
stem  D  D'.  At  F  they  are  united  by  joints  to  the  short  rods 
F  H,  which  are  again  connected  by  joints  at  H  to  a  ring 
which  slides  upon  the  vertical  shaft  D  D'.  From  this  de- 
scription it  will  be  apparent  that  when  the  balls  B  are  drawn 
from  the  axis,  their  upper  arms  E  F  are  caused  to  increase 
their  divergence  in  the  same  manner  as  the  blades  of  scis- 
sors are  opened  by  separating  the  handles.  These,  acting 
upon  the  ring  by  means  of  the  short  links  F  H,  draw  it  down 
the  vertical  axis  from  D  towards  E.  A  contrary  effect  is  pro- 
duced when  the  balls  B  are  brought  closer  to  the  axis,  and 
the  divergence  of  the  rods  B  E  diminished.  A  horizontal 
wheel  W  is  attached  to  the  vertical  axis  D  D',  having  a 
groove  to  receive  a  rope  or  strap  upon  its  rim.  This  strap 
passes  round  the  wheel  or  axis  by  which  motion  is  transmit- 
ted to  the  machinery  to  be  regulated,  so  that  the  spindle  or 
shaft  D  D'  will  always  be  made  to  revolve  with  a  speed  pro- 
portionate to  that  of  the  machinery. 

As  the  shaft  D  D'  revolves,  the  balls  B  are  carried  round 
it  with  a  circular  motion,  and  consequently  acquire  a  centrifu- 
gal force,  which  causes  them  to  recede  from  the  axle,  and 
therefore  to  depress  the  ring  H.  On  the  edge  or  rim  of  this 
ring  is  formed  a  groove,  which  is  embraced  by  the  prongs 
of  a  fork  I,  at  the  extremity  of  one  arm  of  a  lever  whose 
fulcrum  is  at  G.  The  extremity  K  of  the  other  arm  is  con- 
nected by  some  means  with  the  part  of  the  machine  which 
supplies  the  power.  In  the  present  instance,  we  shall  sup- 
pose it  a  steam-engine,  in  which  case  the  rod  K  I  communicates 
with  a  flat  circular  valve  V,  placed  in  the  principal  steam- 
pipe,  and  so  arranged  that,  when  K  is  elevated  as  far  as  by 
their  divergence  the  balls  B  have  power  over  it,  the  passage 
of  the  pipe  will  be  closed  by  the  valve  V,  and  the  passage  of 
steam  entirely  stopped  ;  and  on  the  other  hand,  when  the 
balls  subside  to  their  'lowest  position,  the  valve  will  be  pre- 
sented with  its  edge  in  the  direction  of  the  tube,  so  as  to 
intercept  no  part  of  the  steam. 

The  property  which  renders  this  instrument  so  admirably 
adapted  to  the  purpose  to  which  it  is  applied  is,  that  when 
the  divergence  of  the  balls  is  not  very  considerable,  they 
must  always  revolve  with  the  same  velocity,  whether  they 
move  at  a  greater  or  lesser  distance  from  the  vertical  axis 


CHAP.  XVII.  WATER-REGULATOR.  193 

If  any  circumstance  increases  that  velocity,  the  balls  instantly 
recede  from  the  axis,  and,  closing  the  valve  V,  check  the  sup- 
ply of  steam,  arid  thereby,  diminishing  the  speed  of  the  motion, 
restore  the  machine  to  its  former  rate.  If,  on  the  contrary, 
that  fixed  velocity  be  diminished,  the  centrifugal  force  being 
no  longer  sufficient  to  support  the  balls,  they  descend  towards 
the  axle,  open  the  valve  V,  and,  increasing  the  supply  of  steam, 
restore  the  proper  velocity  of  the  machine. 

When  the  governor  is  applied  to  a  water-wheel,  it  is  made 
to  act  upon  the  shuttle  through  which  the  water  flows,  and 
controls  its  quantity  as  effectually,  and  upon  the  same  prin- 
ciple, as  has  just  been  explained  in  reference  to  the  steam- 
engine.  When  applied  to  a  wind-mill,  it  regulates  the  sail- 
cloth so  as  to  diminish  the  efficacy  of  the  power  upon  the 
arms  as  the  force  of  the  wind  increases,  or  vice  versa. 

In  cases  where  the  resistance  admits  of  easy  and  conve- 
nient change,  the  governor  may  act  so  as  to  accommodate  it  to 
the  varying  energy  of  the  power.  This  is  often  done  in  corn- 
mills,  where  it  acts  upon  the  shuttle  which  metes  out  the 
corn  to  the  millstones.  When  the  power  which  drives  the 
mill  increases,  a  proportionally  increased  feed  of  corn  is  given 
to  the  stones,  so  that,  the  resistance  being  varied  in  the  ratio 
of  the  power,  the  same  velocity  will  be  maintained. 

(304.)  In  some  cases,  the  centrifugal  force  of  the  revolving 
balls  is  not  sufficiently  great  to  control  the  power  or  the  re- 
sistance, and  regulators  of  a  different  kind  must  be  resorted 
to.  The  following  contrivance  is  called  the  water-regula- 
tor :— 

A  common  pump  is  worked  by  the  machine,  whose  motion 
is  to  be  regulated,  and  water  is  thus  raised  and  discharged 
into  a  cistern.  It  is  allowed  to  flow  from  this  cistern  through 
a  pipe  of  a  given  magnitude.  When  thexwater  is  pumped 
up  with  the  same  velocity  as  it  is  discharged  by  this  pipe,  it 
is  evident  that  the  level  of  the  water  in  the  cistern  will  be 
stationary,  since  it  receives  from  the  pump  the  exact  quantity 
which  it  discharges  from  the  pipe.  But  if  the  pump  throw 
in  more  water  in  a  given  time  than  is  discharged  by  the  pipe, 
the  cistern  will  begin  to  be  filled,  and  the  level  of  the  water 
will  rise.  If,  on  the  other  hand,  the  supply  from  the  pump 
be  less  than  the  discharge  from  the  pipe,  the  level  of  the 
water  in  the  cistern  will  subside.  Since  the  rate  at  which 
water  is  supplied  from  the  pump  will  always  be  proportional 
to  the  velocity  of  the  machine,  it  follows  that  every  fluctua- 
17 


194  THE    ELEMENTS   OP  MECHANICS.  CHAP.  XVII. 

tion  in  this  velocity  will  be  indicated  by  the  rising  or  sub- 
siding of  the  level  of  the  water  in  the  cistern,  and  that  level 
never  can  remain  stationary,  except  at  that  exact  velocity 
which  supplies  the  quantity  of  water  discharged  by  the  pipe. 
This  pipe  may  be  constructed  so  as  by  an  adjustment  to  disr 
charge  the  water  at  any  required  rate  ;  and  thus  the  cistern 
may  be  adapted  to  indicate  a  constant  velocity  of  any  pror 
posed  amount. 

If  the  cistern  were  constantly  watched  by  an  attendant^ 
the  velocity  of  the  machine  might  be  abated  by  regulating 
the  power  when  the  level  of  the  water  is  observed  to  rise, 
or  increased  when  it  falls  ;  but  this  is  much  more  effectually 
and  regularly  performed  by  causing  the  surface  of  th.e  water 
itself  to  perform  the  duty.  A  float  or  large  hollow  metal 
ball  is  placed  upon  the  surface  of  the  water  in  the  cistern. 
This  ball  is  connected  with  a  lever  acting  upon  some  part 
of  the  machinery,  which  controls  the  power  or  regulates  the 
amount  of  resistance,  as  already  explained  in  the  case  of 
the  governor.  When  the  level  of  the  water  rises,  the  buoy- 
ancy of  the  ball  causes  it  to  rise  also  with  a  force  equal 
to  the  difference  between  its  own  weight  and  the  weight  of 
as  much  water  as  it  displaces.  By  enlarging  the  floating 
ball,  a  force  may  be  obtained  sufficiently  great  to  move  those 
parts  of  the  machinery  which  act  upon  the  power  or  resist- 
ance, and  thus  either  to  diminish  the  supply  of  the  moving 
principle,  or  to  increase  the  amount  of  the  resistance,  and 
thereby  retard  the  motion  and  reduce  the  velocity  to  its 
proper  limit.  When  the  level  of  the  water  in  the  cistern 
falls,  the  floating  ball,  being  no  longer  supported  on  the 
liquid  surface,  descends  with  the  force  of  its  own  weight, 
and,  producing  an  effect  upon  the  power  or  resistance  contrary 
to  the  former,  increases  the  effective  energy  of  the  one,  or 
diminishes  that  of  the  other,  until  the  velocity  proper  to  the 
machine  be  restored. 

The  sensibility  of  these  regulators  is  increased  by  making 
the  surface  of  water  in  the  cistern  as  small  as  possible ;  for 
then  a  small  change  in  the  rate  at  which  the  water  is  supplied 
by  the  pump  will  produce  a  considerable  change  in  the  level 
of  the  water  in  the  cistern. 

Instead  of  using  a  float,  the  cistern  itself  may  be  suspend- 
ed from  the  lever  which  controls  the  supply  of  the  power, 
and  in  this  case  a  sliding  weight  may  be  placed  on  the  other 
arm,  so  that  it  will  balance  the  cylinder  when  it  contains  that 


CHAP.    XVII. 


REGULATORS.  195 


quantity  of  water  which  corresponds  to  the  fixed  level  already 
explained.  If  the  quantity  of  water  in  the  cistern  be  in- 
creased by  an  undue  velocity  of  the  machine,  the  weight  of 
the  cistern  will  preponderate,  draw  down  the  arm  of  the  lever, 
and  check  the  supply  of  the  power.  If,  on  the  other  hand, 
the  supply  of  water  be  too  small,  the  cistern  will  no  longer 
balance  the  counterpoise,  the  arm  by  which  it  is  suspended 
will  be  raised,  and  the  energy  of  the  power  will  be  increased. 

(305.)  In  the  steam-engine,  the  self-regulating  principle  is 
carried  to  an  astonishing  pitch  of  perfection.  The  machine 
itself  raises  in  a  due  quantity  the  cold  water  necessary  to 
condense  the  steam.  It  pumps  off  the  hot  water  produced  by 
the  steam,  which  has  been  cooled,  and  lodges  it  in  a  reservoir 
for  the  supply  of  the  boiler.  It  carries  from  this  reservoir 
exactly  that  quantity  of  water  which  is  necessary  to  supply 
the  wants  of  the  boiler,  and  lodges  it  therein  according  as  it 
is  required.  It  breathes  the  boiler  of  redundant  steam,  and 
preserves  that  which  remains  fit,  both  in  quantity  and  quality, 
for  the  use  of  the  engine.  It  blows  its  own  fire,  maintaining 
its  intensity,  and  increasing  or  diminishing  it  according  to 
the  quantity  of  steam  which  it  is  necessary  to  raise ;  so  that 
when  much  work  is  expected  from  the  engine,  the  fire  is 
proportionally  brisk  and  vivid.  It  breaks  and  prepares  its 
own  fuel,  and  scatters  it  upon  the  bars  at  proper  times  and 
in  due  quantity.  It  opens  and  closes  its  several  valves  at  the 
proper  moments,  works  its  own  pumps,  turns  its  own  wheels, 
and  is  only  not  alive.  Among  so  many  beautiful  examples 
of  the  self-regulating  principle,  it  is  difficult  to  select.  We 
shall,  however,  mention  one  or  two,  and  for  others  refer  the 
reader  to  that  part  of  the  Cyclopaedia  which  will  contain  a 
detailed  description  of  the  steam-engine  itself. 

It  is  necessary  in  this  machine  that  the  water  in  the  boiler 
be  maintained  constantly  at  the  same  level,  and,  therefore, 
that  as  much  be  supplied,  from  time  to  time,  as  is  consumed 
by  evaporation.  A  pump,  which  is  wrought  by  the  engine 
itself,  supplies  a  cistern  C,.$br-  145.,  with  hot  water.  At  the 
bottom  of  this  cistern  is  a  valve  V  opening  into  a  tube  which 
descends  into  the  boiler.  This  valve  is  connected  by  a  wire 
with  the  arm  of  a  lever  on  the  fulcrum  D,  the  other  arm  E 
of  which  is  also  connected  by  a  wire  with  a  stone  float  F, 
which  is  partially  immersed  in  the  water  of  the  boiler,  and  is 
balanced  by  a  sliding  weight  A.  The  weight  A  only  coun- 
terpoises the  stone  float  F  by  the  aid  of  its  buoyance  in  the 


196  THE    ELEMENTS    OF    MECHANICS  CHAP.    XVII. 

water ;  for  if  the  water  be  removed,  the  stone  F  will  prepon- 
derate, and  raise  the  weight  A.  When  the  water  in  the 
boiler  is  at  its  proper  level,  the  length  of  the  wire  connecting 
the  valve  V  with  the  lever  is  so  adjusted  that  this  valve  shall 
be  closed,  the  wire  at  the  same  time  being  fully  extended. 
When,  by  evaporation,  the  water  in  the  boiler  begins  to  be 
diminished,  the  level  falls,  and  the  stone  weight  F,  being  no 
longer  supported,  overcomes  the  counterpoise  A,  raises  the 
arm  of  the  lever,  and,  pulling  the  wire,  opens  the  valve  V. 
The  water  in  the  cistern  C  then  flows  through  the  tube  into 
the  boiler,  and  continues  to  flow  until  the  level  be  so  raised 
that  the  stone  weight  F  is  again  elevated,  the  valve  V  closed, 
and  the  further  supply  of  water  from  the  cistern  C  suspended. 

In  order  to  render  the  operation  of  this  apparatus  easily 
intelligible,  we  have  here  supposed  an  imperfection  which 
does  not  exist.  According  to  what  has  just  been  stated,  the 
level  of  the  water  in  the  boiler  descends  from  its  proper 
height,  and  subsequently  returns  to  it.  But,  in  fact,  this 
does  not  happen.  The  float  F  and  valve  V  adjust  themselves, 
so  that  a  constant  supply  of  water  passes  through  the  valve, 
which  proceeds  exactly  at  the  same  rate  as  that  at  which  the 
•crater  in  the  boiler  is  consumed. 

(306.)  In  the  same  machine  there  occurs  a  singularly  hap- 
py example  of  self-adjustment,  in  the  method  by  which  the 
strength  of  the  fire  is  regulated.  The  governor  regulates  the 
supply  of  steam  to  the  engine,  and  proportions  it  to  the  work 
to  be  done.  With  this  work,  therefore,  the  demands  upon 
the  boiler  increase  or  diminish,  and  with  these  demands  the 
production  of  steam  in  the  boiler  ought  to  vary.  In  fact,  the 
rate  at  which  steam  is  generated  in  the  boiler  ought  to  be 
equal  to  that  at  which  it  is  consumed  in  the  engine,  other- 
wise one  of  two  effects  must  ensue :  either  the  boiler  will 
fail  to  supply  the  engine  with  steam,  or  steam  will  accumulate 
n  the  boiler,  being  produced  in  undue  quantity,  and,  escaping 
at  the  safety  valve,  will  thus  be  wasted.  It  is,  therefore> 
necessary  to  control  the  agent  which  generates  the  steam, 
namely,  the  fire,  and  to  vary  its  intensity  from  time  to  time, 
proportioning  it  to  the  demands  of  the  engine.  To  accom- 
plish this,  the  following  contrivance  has  been  adopted  : — Let 
T,Jig.  146.,  be  a  tube  inserted  in  the  top  of  the  boiler,  and 
descending  nearly  to  the  bottom.  The  pressure  of  the  steam 
confined  in  the  boiler,  acting  upon  the  surface  of  the  water, 
forces  it  to  a  certain  height  in  the  tube  T.  A  weight  F, 


CHAP.  XVII.  TACHOMETER.  197 

half  immersed  in  the  water  in  the  tube,  is  suspended  by  a 
chain,  which  passes  over  the  wheels  P  1*',  and  is  balanced 
by  a  metal  plate  D,  in  the  same  manner  as  the  stone  float, 
fig.  145.,  is  balanced  by  the  weight  A.  The  plate  D  passes 
through  the  mouth  of  the  flue  E  as  it  issues  finally  from  the 
boiler :  so  that  when  the  plate  D  falls,  it  stops  the  flue,  sus- 
pending thereby  the  draught  of  air  through  the  furnace, 
mitigating  the  intensity  of  the  fire,  and  checking  the  produc- 
tion of  steam.  If,  on  the  contrary,  the  plate  D  be  drawn  up 
the  draught,  is  increased,  the  fire  is  rendered  more  active, 
and  the  production  of  steam  in  the  boiler  is  stimulated. 
Now,  suppose  that  the  boiler  produces  steam  faster  than  the 
engine  consumes  it,  either  because  the  load  on  the  engine 
lias  been  diminished,  and,  therefore,  its  consumption  of 
steam  reduced,  or  because  the  fire  has  become  too  intense ; 
the  consequence  is,  that  the  steam,  beginning  to  accumulate 
in  the  boiler,  will  press  upon  the  surface  of  the  water  with 
increased  force,  and  the  water  will  be  raised  in  the  tube  T. 
The  weight  F  will,  therefore,  be  lifted,  arid  the  plate  D  will 
descend,  diminish,  or  stop  the  draught,  mitigate  the  fire,  and 
retard  the  production  of  steam,  and  will  continue  to  do  so 
until  the  rate  at  which  steam  is  produced  shall  be  commen- 
surate to  the  wants  of  the  engine. 

If,  on  the  other  hand,  the  production  of  steam  be  inade- 
quate to  the  exigency  of  the  machine,  either  because  of  an 
increased  load,  or  of  the  insufficient  force  of  the  fire,  the 
steam  in  the  boiler  will  lose  its  elasticity,  and  the  surface  of 
the  water  not  sustaining  its  wonted  pressure,  the  water  in  the 
tube  T  will  fall ;  consequently  the  weight  F  will  descend, 
and  the  plate  D  will  be  raised.  The  flue  being  thus  opened, 
the  draught  will  be  increased,  and  the  fire  rendered  more  in- 
tense. Thus  the  production  of  steam  becomes  more  rapid, 
and  is  rendered  sufficiently  abundant  for  the  purposes  of  the 
engine.  This  apparatus  is  called  the  self-acting  damper. 

(307.)  When  a  perfectly  uniform  rate  of  motion  has  not 
been  attained,  it  is  often  necessary  to  indicate  small  varia- 
tions of  velocity.  The  following  contrivance,  called  a  ta- 
chometer* has  been  invented  to  accomplish  this.  A  cup, 
Jig.  147.,  is  filled  to  the  level  C  D  with  quicksilver,  and  is 
attached  to  a  spindle,  which  is  whirled  by  the  machine  in  the 
same  manner  as  the  governor  already  described.  It  is  well 

From  the  Greek  words  tachos,  sj>eed;  and  metron,  measure. 


198  THE    ELEMENTS    OF    MECHANICS.          CHAP.    XVII. 

known  that  the  centrifugal  force,  produced  by  this  whirling 
motion,  will  cause  the  mercury  to  recede  from  the  centre,  and 
rise  upon  the  sides  of  the  cup,  so  that  its  surface  will  assume 
the  concave  appearance  represented  in  fg.  148.  In  this 
case,  the  centre  of  the  surface  will  ohviously  have  fallen  be- 
low its  original  level,///,''.  147.,  and  the  edges  will  have  risen 
above  that  level.  As  this  effect  is  produced  by  the  velocity 
of  the  machine,  so  it  is  proportionate  to  that  velocity,  and 
subject  to  corresponding  variations.  Any  method  of  render- 
ing visible  small  changes  in  the  central  level  of  the  surface 
of  the  quicksilver  will  indicate  minute  variations  in  the  ve- 
locity of  the  machine. 

A  glass  tube  A,  open  at  both  ends,  and  expanding  at  one 
extremity  into  a  beil  B,  is  immersed  with  its  wider  end  in  the 
mercury,  the  surface  of  which  will  stand  at  the  same  level 
in  the  bell  B,  and  in  the  cup  C  D.  The  tube  is  so  suspended 
as  to  be  unconnected  with  the  cup.  This  tube  is  then  filled 
to  a  certain  height  A,  with  spirits  tinged  with  some  coloring 
matter,  to  render  it  easily  observable.  When  the  cup  is  whirl- 
ed by  the  machine  to  which  it  is  attached,  the  level  of  the 
quicksilver  in  the  bell  falls,  leaving  more  space  for  the  spirits, 
which,  therefore,  descend  in  the  tube.  As  the  motion  is 
continued,  every  change  of  velocity  causes  a  corresponding 
change  in  the  level  of  the  mercury,  and,  therefore,  also  in 
the  level  A  of  the  spirits.  It  will  be  observed,  that,  in  con- 
sequence of  the  capacity  of  the  bell  B  being  much  greater 
than  that  of  the  tube  A,  a  very  small  change  in  the  level  of 
the  quicksilver  in  the  bell  will  produce  a  considerable  change 
in  the  height  of  the  spirits  in  the  tube.  Thus  this  ingenious 
instrument  becomes  a  very  delicate  indicator  of  variations  in 
the  motion  of  machinery. 

(808.)  The  governor,  and  other  methods  of  regulating  the 
motion  of  machinery  which  have  been  just  described,  are 
adapted  principally  to  cases  in  which  the  proportion  of  the 
resistance  to  the  load  is  subject  to  certain  fluctuations  or 
gradual  changes,  or  at  least  to  cases  in  which  the  resistance 
is  not  at  any  time  entirely  withdrawn,  nor  the  energy  of  the 
power  actually  suspended.  Circumstances,  however,  frequent- 
ly occur  in  which,  while  the  power  remains  in  full  activity, 
the  resistance  is  at  intervals  suddenly  removed,  and  as 
suddenly  again  returns.  On  the  other  hand,  cases  also  pre- 
sent themselves,  in  which,  while  the  resistance  is  continued, 
the  impelling  power  is  subject  to  intermission  at  regular  pe- 


UHAP.    XVII.  ACCUMULATION    OF    FORCE.  199 

riods.  In  the  former  case,  the  machine  would  be  driven 
with  a  ruinous  rapidity  during  those  periods  at  which  it  is 
relieved  from  its  load,  and,  on  the  return  of  the  load,  every 
part  would  suffer  a  violent  strain,  from  its  endeavor  to  re- 
tain the  velocity  which  it  had  acquired,  and  the  speedy  de- 
struction of  the  engine  could  not  fail  to  ensue.  In  the 
latter  case,  the  motion  would  be  greatly  retarded  or  entirely 
suspended  during  those  periods  at  which  the  moving  power 
is  deprived  of  its  activity,  and,  consequently,  the  motion  which 
it  would  communicate  would  be  so  irregular  as  to  be  useless 
for  the  purposes  of  manufactures. 

It  is  also  frequently  desirable,  by  means  of  a  weak  but 
continued  power,  to  produce  a  severe  but  instantaneous  effect. 
Thus  a  blow  may  be  required  to  be  given  by  the  muscular 
action  of  a  man's  arm  with  a  force  to  which,  unaided  by 
mechanical  contrivance,  its  strength  would  be  entirely  in- 
adequate. 

In  all  these  cases,  it  is  evident  that  the  object  to  be  at- 
tained is,  an  effectual  method  of  accumulating  the  energy  of 
the  power,  so  as  to  make  it  available  after  the  action  by  which 
it  has  been  produced  has  ceased.  Thus,  in  the  case  in  which 
the  load  is  at  periodical  intervals  withdrawn  from  the  machine, 
if  the  force  of  the  power  could  be  imparted  to  something  by 
which  it  would  be  preserved,  so  as  to  be  brought  against  the 
load  when  it  again  returned,  the  inconvenience  would  be 
removed.  In  like  manner,  in  the  case  where  the  power 
itself  is  subject  to  intermission,  if  a  part  of  the  force  which  it 
exerts  in  its  intervals  of  action  could  be  accumulated  and 
preserved,  it  might  be  brought  to  bear  upon  the  machine 
during  its  periods  of  suspension.  By  the  same  means  of  ac- 
cumulating force,  the  strength  of  an  infant,  by  repeated 
efforts,  might  produce  effects  which  would  be  vainly  attempt- 
ed by  the  single  and  momentary  action  of  the  strongest  man. 

(309.)  The  property  of  inertia,  explained  and  illustrated 
in  the  third  and  fourth  chapters  of  this  volume,  furnishes  an 
easy  and  effectual  method  of  accomplishing  this.  A  mass 
of  matter  retains,  by  virtue  of  its  inertia,  the  whole  of  any 
force  which  may  have  been  given  to  it,  except  that  part  of 
which  friction  and  the  atmospheric  resistance  deprive  it. 
By  contrivances  which  are  well  known,  and  present  no  diffi- 
culty, the  part  of  the  moving  force  thus  lost  may  be  rendered 
comparatively  small,  and  the  moving  mass  rnay  be  regarded 
as  retaining  nearly  the  whole  of  the  force  impressed  upon  it. 


200  THE  ELEMENTS  OF  MECHANICS.  CHAP.  XVII. 

To  render  this  method  of  accumulating  force  fully  intelligi- 
ble, let  us  first  imagine  a  polished  level  plane  on  which  a 
heavy  globe  of  metal,  also  polished,  is  placed.  It  is  evident 
that  the  globe  will  remain  at  rest  on  any  part  of  the  plane 
without  a  tendency  to  move  in  any  direction.  As  the  friction 
is  nearly  removed  by  the  polish  of  the  surfaces,  the  globe 
will  be  easily  moved  by  the  least  force  applied  to  it.  Sup- 
pose a  slight  impulse  given  to  it,  which  will  cause  it  to  move 
at  the  rate  of  one  foot  in  a  second.  Setting  aside  the  effects 
of  friction,  it  will  continue  to  move  at  this  rate  for  any  length 
of  time.  The  same  impulse  repeated  will  increase  its  speed 
to  two  feet  per  second;  a  third  impulse  to  three  feet;  and  so 
on.  Thus  10,000  repetitions  of  the  impulse  will  cause  it 
to  move  at  the  rate  of  10,000  feet  per  second.  If  the  body 
to  which  these  impulses  were  communicated  were  a  cannon 
ball,  it  might,  by  a  constant  repetition  of  the  impelling  force, 
be  at  length  made  to  move  with  as  much  force  as  if  it  were 
projected  from  the  most  powerful  piece  of  ordnance.  The 
force  with  which  the  ball  in  such  a  case  would  strike  a  build- 
ing might  be  sufficient  to  reduce  it  to  ruins,  and  yet  such 
force  would  be  nothing  more  than  the  accumulation  of  a 
number  of  weak  efforts  not  beyond  the  power  of  a  child  to 
exert,  which  are  stored  up,  and  preserved,  as  it  were,  by  the 
moving  mass,  and  thereby  brought  to  bear,  at  the  same  mo- 
ment, upon  the  point  to  which  the  force  is  directed.  It  is 
the  sum  of  a  number  of  actions  exerted  successively,  and, 
during  a  long  interval,  brought  into  operation  at  one  and  the 
same  moment. 

But  the  case  which  is  here  supposed  cannot  actually  occur ; 
because  we  have  not  usually  any  practical  means  of  moving 
a  body  for  any  considerable  time  in  the  same  direction  with- 
out much  friction,  and  without  encountering  numerous  ob- 
stacles which  would  impede  its  progress.  It  is  not,  however, 
essential  to  the  effect  which  is  to  be  produced,  that  the  mo- 
tion should  be  in  a  straight  line.  If  a  leaden  weight  be  at- 
tached to  the  end  of  a  light  rod  or  cord,  and  be  whirled  by 
the  force  of  the  arm  in  a  circle,  it  will  gradually  acquire  in- 
creased speed  and  force,  and  at  length  may  receive  an  impetus 
which  would  cause  it  to  penetrate  a  piece  of  board  as  effect- 
ually as  if  it  were  discharged  from  a  musket. 

The  force  of  a  hammer  or  sledge  depends  partly  on  its 
weight,  but  much  more  on  the  principle  just  explained.  Were 
it  allowed  merely  to  fall  by  the  force  of  its  weight  upon  the 


CHAP.    XVII.  FLV-VVHEEL.  201 

head  of  a  nail,  or  upon  a  bar  of  heated  iron  which  is  to  be 
flattened,  an  inconsiderable  effect  would  be  produced.  But 
when  it  is  wielded  by  the  arm  of  a  man,  it  receives  at  every 
moment  of  its  motion  increased  force,  which  is  finally  ex- 
pended in  a  single  instant  on  the  head  of  the  nail,  or  on  the 
bar  of  iron. 

The  effects  of  flails  in  threshing,  of  clubs,  whips,  canes, 
and  instruments  for  striking,  axes,  hatchets,  cleavers,  and  all 
instruments  which  cut  by  a  blow,  depend  on  the  same  prin- 
ciple, and  are  similarly  explained. 

The  bow-string  which  impels  the  arrow  does  not  produce 
its  effect  at  once.  It  continues  to  act  upon  the  shaft  until  it 
resumes  its  straight  position,  and  then  the  arrow  takes  flight 
with  the  force  accumulated  during  the  continuance  of  the 
action  of  the  string,  from  the  moment  it  was  disengaged  from 
the  finger  of  the  bow-man. 

Fire-arms  themselves  act  upon  a  similar  principle,  as  also 
the  air-gun  and  steam-gun.  In  these  instruments,  the  ball 
is  placed  in  a  tube,  and  suddenly  exposed  to  the  pressure  of 
a  highly  elastic  fluid,  either  produced  by  explosion  as  in  fire- 
arms, by  previous  condensation  as  in  the  air-gun,  or  by  the 
evaporation  of  highly  heated  liquids  as  in  the  steam-gun. 
But  in  every  case  this  pressure  continues  to  act  upon  it  until 
it  leaves  the  mouth  of  the  tube,  and  then  it  departs  with  the 
whole  force  communicated  to  it  during  its  passage  along  the 
tube. 

(310.)  From  all  these  considerations  it  will  easily  be  per- 
ceived that  a  mass  of  inert  matter  may  be  regarded  as  a 
magazine  in  which  force  may  be  deposited  and  accumulated, 
to  be  used  in  any  way  which  may  be  necessary.  For  many 
reasons,  which  will  be  sufficiently  obvious,  the  form  common- 
ly given  to  the  mass  of  matter  used  for  this  purpose  in  ma- 
chinery, is  that  of  a  wheel,  in  the  rim  of  which  it  is  principal- 
ly collected.  Conceive  a  massive  ring  of  metal,  Jig.  149., 
connected  with  a  central  box  or  nave  by  light  spokes,  and 
turning  on  an  axis  with  little  friction.  Such  an  apparatus  is 
called  a  fly-wheel.  If  any  force  be  applied  to  it,  with  that 
force  (making  some  slight  deduction  for  friction)  it  will  move, 
and  will  continue  to  move  until  some  obstacle  be  opposed  to 
its  motion,  which  will  receive  from  it  a  part  of  the  force  it 
has  acquired.  The  uses  of  this  apparatus  will  be  easily  un- 
derstood by  examples  of  its  application. 

Suppose  that  a  heavy  stamper  or  hammer  is  to  be  raised  to 


202  THE    ELEMENTS    OF    MECHANICS.  CHAP.  XVII. 

a  certain  height,  and  thence  to  be  allowed  to  fall,  and  that 
the  power  used  for  this  purpose  is  a  water-wheel.  While  the 
stamper  ascends,  the  power  of  the  wheel  is  nearly  balanced 
by  its  weight,  and  the  motion  of  the  machine  is  slow.  But 
the  moment  the  stamper  is  disengaged  and  allowed  to  fall,  the 
power  of  the  wheel,  having  no  resistance,  nor  any  object  on 
which  to  expend  itself,  suddenly  accelerates  the  machine, 
which  moves  with  a  speed  proportioned  to  the  amount  of  the 
power,  until  it  again  engages  the  stamper,  when  its  velocity 
is  as  suddenly  checked.  Every  part  suffers  a  strain,  and  the 
machine  moves  again  slowly  until  it  discharges  its  load,  when 
it  is  again  accelerated,  and  so  on.  In  this  case,  besides  the 
certainty  of  injury  and  wear,  and  the  probability  of  fracture 
from  the  sudden  and  frequent  changes  of  velocity,  nearly  the 
whole  force  exerted  by  the  power  in  the  intervals  between  the 
commencement  of  each  descent  of  the  stamper  and  the  next 
ascent,  is  lost.  These  defects  are  removed  by  a  fly-wheel. 
When  the  stamper  is  discharged,  the  energy  of  the  power  is 
expended  in  moving  the  wheel,  which,  by  reason  of  its  great 
mass,  will  not  receive  an  undue  velocity.  In  the  interval  be- 
tween the  descent  and  ascent  of  the  stamper,  the  force  of 
the  power  is  lodged  in  the  heavy  rim  of  the  fly-wheel.  When 
the  stamper  is  again  taken  up  by  the  machine,  this  force  is 
brought  to  bear  upon  it,  combined  with  the  immediate  power 
of  the  water-wheel,  and  the  stamper  is  elevated  with  nearly 
the  same  velocity  as  that  with  which  the  machine  moved  in 
the  interval  of  its  descent. 

(311.)  In  many  cases,  when  the  moving  power  is  not  sub- 
ject to  variation,  the  efficacy  of  the  machine  to  transmit  it 
to  the  working  point  is  subject  to  continual  change.  The 
several  parts  of  every  machine  have  certain  periods  of  motion, 
in  which  they  pass  through  a  variety  of  positions,  to  which 
they  continually  return  after  stated  intervals.  In  these  differ- 
ent positions,  the  effect  of  the  power  transmitted  to  the  work- 
ing point  is  different ;  and  cases  even  occur  in  which  this  ef- 
fect is  altogether  annihilated,  and  the  machine  is  brought  into 
a  predicament  in  which  the  power  loses  all  influence  over  the 
weight.  In  such  cases,  the  aid  of  a  fly-wheel  is  effectual  and 
indispensable.  In  those  phases  of  the  machine,  which  are 
most  favorable  to  the  transmission  of  force,  the  fly-wheel 
shares  the  effect  of  the  power  with  the  load,  and,  retaining  the 
force  thus  received,  directs  it  upon  the  load  at  the  moments 
when  the  transmission  of  power  by  the  machine  is  either  fee 


CHAP.  XVII.  FLY-WHEEL    AND    CRANK.  203 

ble  or  altogether  suspended.  These  general  observations 
will,  perhaps,  be  more  clearly  apprehended  by  an  example  of 
an  application  of  the  fly-wheel,  in  a  case  such  as  those  now 
alluded  to. 

Let  A  B  C  D  E  F,fg.  150.,  be  a  crank,  which  is  a  double 
winch  (  (252.)  and  Jig.  89.),  by  which  an  axle,  A  B  E  F,  is 
,to  be  turned.  Attached  to  the  middle  of  C  D  by  a  joint  is  a 
rod,  which  is  connected  with  a  beam,  worked  with  an  alter- 
jiate  motion  on  a  centre,  like  the  brake  of  a  pump,  and  driven 
joy  any  constant  power,  such  as  a  steam-engine.  The  bar 
G  D  is  to  be  carried  with  a  circular  motion  round  the  axis 
4-  E.  Let  the  machine,  viewed  in  the  direction  A  B  E  F  of 
the  axis,  be  conceived  to  be  represented  in  fig.  151.,  where 
A  represents  the  centre  round  which  the  motion  is  to  be  pro- 
duced, and  G  the  point  where  the  connecting  rod  G  H  is  at- 
tached to  the  arm  of  the  crank.  The  circle  through  which 
G  is  to  be  urged  by  the  rod  is  represented  by  the  dotted  line. 
In  the  position  represented  in  jig.  151.,  the  rod  acting  in  the 
direction  H  G  has  its  full  power  to  turn  the  crank  G  A  round 
the  centre  A.  As  the  crank  comes  into  the  position  repre- 
sented in  Jig.  152.,  this  power  is  diminished,  and  when  the 
point  G  comes  immediately  below  A,  as  in  Jig.  153.,  the  force 
in  the  direction  H  G  has  no  effect  in  turning  the  crank  round 
A,  but,  on  the  contrary,  is  entirely  expended  in  pulling  the 
crank  in  the  direction  A  G,  and,  therefore,  only  acts  upon  the 
pivots  or  gudgeons  which  support  the  axle.  At  this  crisis  of 
the  motion,  therefore,  the  whole  effective  energy  of  the  power 
is  annihilated. 

After  the  crank  has  passed  to  the  position  represented  in 
Jig.  154.,  the  direction  of  the  force  which  acts  upon  the  con- 
necting rod  is  changed,  and  now  the  crank  is  drawn  upward 
in  the  direction  G  H.  In  this  position,  the  moving  force  has 
some  efficacy  to  produce  rotation  round  A,  which  efficacy 
continually  increases  until  the  crank  attains  the  position  shown 
in  Jig.  155.,  when  its  power  is  greatest.  Passing  from  this 
position,  its  efficacy  is  continually  diminished,  until  the  point 
G  comes  immediately  above  the  axis  A,  Jig.  156.  Here  again 
the  power  loses  all  its  efficacy  to  turn  the  axle.  The  force 
in  the  direction  G  H  or  H  G  can  obviously  produce  no  other 
effect  than  a  strain  upon  the  pivots  or  gudgeons. 

In  the  critical  situations  represented  in  Jig.  153.  and  fig. 
156.,  the  machine  would  be  incapable  of  moving,  were  the 
immediate  force  of  the  power  the  only  impelling  principle 


204  THE  ELEMENTS  OF  MECHANICS.  CHAP.  XVII. 

But,  having  been  previously  in  motion  by  virtue  of  the  inertia 
of  its  various  parts,  it  has  a  tendency  to  continue  in  motion  ; 
and  if  the  resistance  of  the  load  and  the  effects  oi  iriction  be 
not  too  great,  this  disposition  to  preserve  its  state  of  motion 
will  extricate  the  machine  from  the  dilemma  in  which  it  is  in- 
volved, in  the  cases  just  mentioned,  by  the  peculiar  arrange- 
ment of  its  parts.  In  many  cases,  however,  the  force  thus 
acquired  during  the  phases  of  the  machine  in  which  the  pow- 
er is  active,  is  insufficient  to  carry  it  through  the  dead  points 
(fig-  153.  and  Jig.  156.);  and  in  all  cases,  the  motion  would 
be  very  unequal,  being  continually  retarded  as  it  approached 
these  points,  and  continually  accelerated  after  it  passed  them. 
A  fly-wheel  attached  to  the  axis  A,  or  to  some  other  part  of 
the  machinery,  will  effectually  remove  this  defect.  When 
the  crank  assumes  the  positions  in  Jig.  151.  andj#g\  155.,  the 
power  is  in  full  play  upon  it,  and  a  share  of  the  effect  is  im- 
parted to  the  massive  rim  of  the  fly-wheel.  When  the  crank 
gets  into  the  predicament  exhibited  \njig.  153.  and  j£g\  156., 
the  momentum,  which  the  fly-wheel  received  when  the  crank 
acted  with  most  advantage,  immediately  extricates  the  ma- 
chine, and,  carrying  the  crank  beyond  the  dead  point,  brings 
the  power  again  to  bear  upon  it. 

The  astonishing  effects  of  a  fly-wheel,  as  an  accumulator 
of  force,  have  led  some  into  the  error  of  supposing  that  such 
an  apparatus  increases  the  actual  power  of  a  machine.  It  is 
hoped,  however,  that  after  what  has  been  explained  respect- 
ing the  inertia  of  matter  and  the  true  effects  of  machines,  the 
reader  will  not  be  liable  to  a  similar  mistake.  On  the  con- 
trary, as  a  fly  cannot  act  without  friction,  and  as  the  amount 
of  the  friction,  like  that  of  inertia,  is  in  proportion  to  the 
weight,  a  portion  of  the  actual  moving  force  must  unavoida- 
bly be  lost  by  the  use  of  a  fly.  In  cases,  however,  where  a 
fly  is  properly  applied,  this  loss  of  power  is  inconsiderable, 
compared  with  the  advantageous  distribution  of  what  re- 
mains. 

As  an  accumulator  of  force,  a  fly  can  never  have  more  force 
than  has  been  applied  to  put  it  in  motion.  In  this  respect  it 
is  analogous  to  an  elastic  spring,  or  the  force  of  condensed 
air,  or  any  other  power  which  derives  its  existence  from  causes 
purely  mechanical.  In  bending  a  spring,  a  gradual  expendi- 
ture of  power  is  necessary.  On  the  recoil,  this  power  is  ex- 
erted in  a  much  shorter  time  than  that  consumed  in  its  pro- 
duction, but  its  total  amount  is  not  altered.  Air  is  condens- 


CHAP.  XVII.  FLY-WHEEL.  205 

ed  by  a  succession  of  manual  efforts,  one  of  which  alone 
would  be  incapable  of  projecting  a  leaden  ball  with  any  con- 
siderable force,  and  all  of  which  could  not  be  immediately 
applied  to  the  ball  at  the  same  instant.  But  the  reservoir  of 
condensed  air  is  a  magazine  in  which  a  great  number  of  such 
efforts  are  stored  up,  so  as  to  be  brought  at  once  into  action. 
If  a  ball  be  exposed  to  their  effect,  it  may  be  projected  with 
a  destructive  force. 

In  mills  for  rolling  metal,  the  fly-wheel  is  used  in  this  way. 
The  water-wheel  or  other  moving  power  is  allowed  for  some 
time  to  act  upon  the  fly-wheel  alone,  no  load  being  placed 
upon  the  machine.  A  force  is  thus  gained  which  is  sufficient 
to  roll  a  large  piece  of  metal,  to  which  without  such  means 
the  mill  would  be  quite  inadequate.  In  the  same  manner  a 
force  may  be  gained  by  the  arm  of  a  man  acting  on  a  fly  for 
a  few  seconds,  sufficient  to  impress  an  image  on  a  piece  of 
metal  by  an  instantaneous  stroke.  The  fly  is,  therefore,  the 
principal  agent  in  coining  presses. 

(312.)  The  power  of  a  fly  is  often  transmitted  to  the  work- 
ing point  by  means  of  a  screw.  At  the  extremities  of  the 
cross  arm  A  B,  jig.  157.,  which  works  the  screw,  two  heavy 
balls  of  metal  are  placed.  When  the  arm  A  B  is  whirled 
round,  those  masses  of  metal  acquire  a  momentum,  by  which 
the  screw,  being  driven  downwards,  urges  the  die  with  an  im- 
mense force  against  the  substance  destined  to  receive  the  im- 
pression. 

Some  engines  used  in  coining  have  flies  with  arms  four  feet 
long,  bearing  one  hundred  weight  at  each  of  their  extremities. 
By  turning  such  an  arm  at  the  rate  of  one  entire  circumfer- 
ence in  a  second,  the  die  will  be  driven  against  the  metal  with 
the  same  force  as  that  with  which  7500  pounds  weight  would 
fall  from  the  height  of  16  feet ;  an  enormous  power,  if  the 
simplicity  and  compactness  of  the  machine  be  considered. 

The  place  to  be  assigned  to  a  fly-wheel  relatively  to  the 
other  parts  of  the  machinery  is  determined  by  the  purpose  for 
which  it  is  used.  If  it  be  intended  to  equalize  the  action,  it 
should  be  near  the  working  point.  Thus,  in  a  steam-engine, 
it  is  placed  on  the  crank  which  turns  the  axle  by  which  the 
power  of  the  engine  is  transmitted  to  the  object  it  is  finally 
designed  to  affect.  On  the  contrary,  in  handmills,  such 
as  those  commonly  used  for  grinding  coffee,  &,c.,  it  is 
placed  upon  the  axis  of  the  winch  by  which  the  machine  is 
worked. 

18 


206  THE    ELEMENTS  OF  MECHANICS.  CHAP.  XVIII. 

The  open  work  of  fenders,  fire-grates,  and  similar  orna- 
mental articles  constructed  in  metal,  is  produced  by  the  action 
of  a  fly,  in  the  manner  already  described.  The  cutting  tool, 
shaped  according  to  the  pattern  to  be  executed,  is  attached  to 
the  end  of  the  screw ;  and  the  metal  being  held  in  a  proper  po- 
sition beneath  it,  the  fly  is  made  to  urge  the  tool  downwards 
with  such  force  ag  to  stamp  out  pieces  of  the  required  figure. 
When  the  pattern  is  complicated,  and  it  is  necessary  to  pre- 
serve with  exactness  the  relative  situation  of  its  different  parts, 
a  number  of  punches  are  impelled  together,  so  as  to  strike 
the  entire  piece  of  metal  at  the  same  instant,  and  in  this  man- 
ner the  most  elaborate  open  work  is  executed  by  a  single  stroke 
of  the  hand, 


CHAPTER  XVIII 

MECHANICAL    CONTRIVANCES    FOR    MODIFYING    MOTION. 

313.)  THE  classes  of  simple  machines  denominated  me- 
chanic powers,  have  relation  chiefly  to  the  peculiar  principle 
which  determines  the  action  of  the  power  on  the  weight  or 
resistance.  In  explaining  this  arrangement,  various  other 
reflections  have  been  incidentally  mixed  up  with  our  investi- 
gations :  yet  still  much  remains  to  be  unfolded  before  the 
student  can  form  a  just  notion  of  those  means,  by  which  the 
complex  machinery  used  in  the  arts  and  manufactures  so  ef- 
fectually attains  the  ends,  to  the  accomplishment  of  which  it 
is  directed, 

By  a  power  of  a  given  energy  to  oppose  a  resistance  of  a 
different  energy,  or  by  a  moving  principle  having  a  given 
velocity  to  generate  another  velocity  of  a  different  amount,  is 
only  one  of  the  many  objects  to  be  effected  by  a  machine.  In 
the  arts  and  manufactures,  the  kind  of  motion  produced  is  gen- 
erally of  greater  importance  than  its  rate.  The  latter  may  ef- 
fect the  quantity  of  work  done  in  a  given  time,  but  the  former 
is  essential  to  the  performance  of  the  work  in  any  quantity  what- 
ever. In  the  practical  application  of  machines,  the  object  to  be 
attained  is  generally  to  communicate  to  the  working  point  some 
peculiar  sort  of  motion  suitable  to  the  uses  for  which  the  ma- 
chine is  intended ;  but  it  rarely  happens  that  the  moving  power 
has  this  sort  of  motion.  Hence  the  machine  must  be  so  con- 


CHAP.  XVIII.  MODIFICATION  Of  MOTION.  207 

trived  that,  while  that  part  on  which  this  power  acts  is  capa- 
ble of  moving  in  obedience  to  it,  its  connection  with  the 
other  parts  shall  be  such  that  the  working  point  may  receive 
that  motion  which  is  necessary  for  the  purposes  to  which  the 
machine  is  applied. 

To  give  a  perfect  solution  of  this  problem,  it  would  be 
necessary  to  explain,  first,  all  the  varieties  of  moving  powers 
which  are  at  our  disposal ;  secondly,  all  the  variety  of  mo- 
tions which  it  may  be  necessary  to  produce ;  and,  thirdly 
to  show  all  the  methods  by  which  each  variety  of  prime 
mover  may  be  made  to  produce  the  several  species  of  motion 
in  the  working  point.  It  is  obvious  that  such  an  enumeration 
would  be  impracticable,  and  even  an  approximation  to  it 
would  be  unsuitable  to  the  present  treatise.  Nevertheless, 
so  much  ingenuity  has  been  displayed  in  many  of  the  con- 
trivances for  modifying  motion,  and  an  acquaintance  with 
some  of  them  is  so  essential  to  a  clear  comprehension  of  the 
nature  and  operation  of  complex  machines,  that  it  would  be 
improper  to  omit  some  account  of  those  at  least  which  most 
frequently  occur  in  machinery,  or  which  are  most  conspicuous 
for  elegance  and  simplicity. 

The  varieties  of  motion  which  most  commonly  present 
themselves  in  the  practical  application  of  mechanics  may  be 
divided  into  rectilinear  and  rotatory.  In  rectilinear  motion 
the  several  parts  of  the  moving  body  proceed  in  parallel 
straight  lines  with  the  same  speed.  In  rotatory  motion  the 
several  points  revolve  round  an  axis,  each  performing  a  com- 
plete circle,  or  similar  parts  of  a  circle,  in  the  same  time. 

Each  of  these  may  again  be  resolved  into  continued  and 
reciprocating.  In  a  continued  motion,  whether  rectilinear 
or  rotatory,  the  parts  move  constantly  in  the  same  direction, 
whether  that  be  in  parallel  straight  lines,  or  in  rotation  on  an 
axis.  In  reciprocating  motion  the  several  parts  move  alter- 
nately in  opposite  directions,  tracing  the  same  spaces  from 
end  to  end  continually.  Thus  there  are  four  principal  species 
of  motion  which  more  frequently  than  any  others  act  upon, 
or  are  required  to  be  transmitted  by,  machines  : — 

1.  Continued  rectilinear  motion. 

2.  Reciprocating  rectilinear  motion. 

3.  Continued  circular  motion. 

4.  Reciprocating  circular  motion. 

These  will  be  more  clearly  understood  by  examples  of  each 
kind. 


208  THE  ELEMENTS  OF  MECHANICS.         CHAP.  XVIII 

Continued  rectilinear  motion  is  observed  in  the  flowing 
of  a  river,  in  a  fall  of  water,  in  the  blowing  of  the  wind,  in 
the  motion  of  an  animal  upon  a  straight  road,  in  the  perpen- 
dicular fall  of  a  heavy  body,  in  the  motion  of  a  body  down 
an  inclined  plane. 

Reciprocating  rectilinear  motion  is  seen  in  the  piston  of  a 
common  syringe,  in  the  rod  of  a  common  pump,  in  the  ham- 
mer of  a  pavier,  the  piston  of  a  steam-engine,  the  stampers 
of  a  fulling-mill. 

Continued  circular  motion  is  exhibited  in  all  kinds  of 
wheel-work,  and  is  so  common,  that  to  particularize  it  is 
needless. 

Reciprocating  circular  motion  is  seen  in  the  pendulum  of  a 
clock,  and  in  the  balance-wheel  of  a  watch. 

We  shall  now  explain  some  of  the  contrivances  by  which 
a  power  having  one  of  these  motions  may  be  made  to  com- 
municate either  the  same  species  of  motion  changed  in  its 
velocity  or  direction,  or  any  of  the  other  three  kinds  of  mo- 
tion. 

(314.)  By  a  continued  rectilinear  motion  another  continu- 
ed rectilinear  motion  in  a  different  direction  may  be  produced, 
by  one  or  more  fixed  pulleys.  A  cord  passed  over  these,  one 
end  of  it  being  moved  by  the  power,  will  transmit  the  same 
motion  unchanged  to  the  other  end.  If  the  directions  of  the 
two  motions  cross  each  other,  one  fixed^pulley  will  be  suffi- 
cient ;  see  jig.  113.,  where  the  hand  takes  the  direction  of 
the  one  motion,  and  the  weight  that  of  the  other.  In  this  case, 
the  pulley  must  be  placed  in  the  angle  at  which  the  directions 
of  the  two  motions  cross  each  other.  If  this  angle  be  distant 
from  the  places  at  which  the  objects  in  motion  are  situate, 
an  inconvenient  length  of  rope  may  be  necessary.  In  this 
case,  the  same  may  be  effected  by  the  use  of  two  pulleys,  as 
\njig.  158. 

If  the  directions  of  the  two  motions  be  parallel,  two  pul- 
leys must  be  used,  as  in  Jig.  158.,  where  P'  A'  is  one  motion, 
and  B  W  the  other.  In  these  cases,  the  axles  of  the  two 
wheels  are  parallel. 

It  may  so  happen  that  the  directions  of  the  two  motions 
neither  cross  each  other  nor  are  parallel.  This  would  hap- 
pen, for  example,  if  the  direction  of  one  were  upon  the  paper 
in  the  line  P  A,  while  the  other  were  perpendicular  to  the 
paper  from  the  point  O.  In  this  case,  two  pulleys  should  be 
used,  the  axle  of  one  O'  being  perpendicular  to  the  paper, 


CHAP.  XVIII.  MODIFICATION    OF    MOTION.  209 

while  the  axle  of  the  other  O  should  be  on  the  paper.  This 
will  be  evident  by  a  little  reflection. 

In  general,  the  axle  of  each  pulley  must  be  perpendicular 
to  the  two  directions  in  which  the  rope  passes  from  its  groove ; 
and  by  due  attention  to  this  condition  it  will  be  perceived, 
that  a  continued  rectilinear  motion  may  be  transferred  from 
any  one  direction  to  any  other  direction,  by  means  of  a  cord 
and  two  pulleys,  without  changing  its  velocity. 

If  it  be  necessary  to  change  the  velocity,  any  of  the  sys- 
tems of  pulleys  described  in  Chapter  XV.  may  be  used  in  ad- 
dition to  the  fixed  pulleys. 

By  the  wheel  and  axle  any  one  continued  rectilinear  motion 
may,  be  made  to  produce  another  in  any  other  direction,  and 
with  any  other  velocity.  It  has  been  already  explained  (250.) 
that  the  proportion  of  the  velocity  of  the  power  to  that  of  the 
weight  is  as  the  diameter  of  the  wheel  to  the  diameter  of  the 
axle.  The  thickness  of  the  axle  being  therefore  regulated  in 
relation  to  the  size  of  the  wheel,  so  that  their  diameters  shall 
have  that  proportion  which  subsists  between  the  proposed  ve- 
locities, one  condition  of  the  problem  will  be  fulfilled.  The 
rope  coiled  upon  the  axle  may  be  carried,  by  means  of  one  or 
more  fixed  pulleys,  into  the  direction  of  one  of  the  proposed 
motions,  while  that  which  surrounds  the  wheel  is  carried  into 
the  direction  of  the  other  by  similar  means. 

(315.)  By  the  wheel  and  axle  a  continued  rectilinear  mo- 
tion may  be  made  to  produce  a  continued  rotatory  motion,  or 
vice  versa.  If  the  power  be  applied  by  a  rope  coiled  upon 
the  wheel,  the  continued  motion  of  the  power  in  a  straight 
line  will  cause  the  machine  to  have  a  rotatory  motion.  Again, 
if  the  weight  be  applied  by  a  rope  coiled  upon  the  axle,  a 
power  having  a  rotatory  motion  applied  to  the  wheel  will 
cause  the  continued  ascent  of  the  weight  in  a  straight  line. 

Continued  rectilinear  and  rotatory  motions  may  be  made  to 
produce  each  other,  by  causing  a  toothed  wheel  to  work  in  a 
straight  bar,  called  a  rack,  carrying  teeth  upon  its  edge. 
Such  an  apparatus  is  represented  in  Jig.  159. 

In  some  cases,  the  teeth  of  the  wheel-work  in  the  links  of  a 
chain.  The  wheel  is  then  called  a  rag-wheel,  Jig.  160. 

Straps,  bands,  or  ropes,  may  communicate  rotation  to  a 
wheel,  by  their  friction  in  a  groove  upon  its  edge. 

A  continued  rectilinear  motion  is  produced  by  a  continued 
circular  motion  in  the  case  of  a  screw.  The  lever  which 
turns  the  screw  has  a  continued  circular  motion,  while  the 
18  * 


. — f. 


210  TIIE    ELEMENTS    OF    MECHANICS.         CHAP.    XVIII. 

screw  itself  advances  with  a  continued  rectilinear  mo- 
tion. 

The  continued  rectilinear  motion  of  a  stream  of  water  act- 
ing upon  a  wheel  produces  continued  circular  motion  in  the 
wheel,  Jig.  93,  94, 95.  In  like  manner  the  continued  rectilin- 
ear motion  of  the  wind  produces  a  continued  circular  mo- 
tion in  the  arms  of  a  windmill. 

Cranes  for  raising  and  lowering  heavy  weights  convert  a 
circular  motion  of  the  power  into  a  continued  rectilinear  mo- 
tion of  the  weight. 

(310.)  Continued  circular  motion  may  produce  reciprocat- 
ing rectilinear  motion,  by  a  great  variety  of  ingenious  con- 
trivances. 

Reciprocating  rectilinear  motion  is  used  when  heavy  stamp- 
ers are  to  be  raised  to  a  certain  height,  and  allowed  to  fal! 
upon  some  object  placed  beneath  them.  This  may  be  accom- 
plished by  a  wheel  bearing  on  its  edge  curved  teeth,  called 
wipers.  The  stamper  is  furnished  with  a  projecting  arm  or 
peg,  beneath  which  the  wipers  are  successively  brought  by 
the  revolution  of  the  wheel.  As  the  wheel  revolves,  the  wiper 
raises  the  stamper,  until  its  extremity  passes,  the  extremity  of 
the  projecting  arm  of  the  stamper,  when  the  latter  immediate- 
ly falls  by  its  own  weight.  It  is  then  taken  up  by  the  next 
wiper,  and  so  the  process  is  continued. 

A  similar  effect  is  produced  if  the  wheel  be  partially  fur- 
nished with  teeth,  and  the  stamper  carry  a  rack  in  which 
these  teeth  work.  Such  an  apparatus  is  represented  in  Jig. 
161. 

It  is  sometimes  necessary  that  the  reciprocating  rectilinear 
motion  shall  be  performed  at  a  certain  varying  rate  in  both 
directions.  This  may  be  accomplished  by  the  machine  rep- 
resented in  Jig.  16*2.  A  wheel  upon  the  axle  C  turns  uniform- 
ly in  the  direction  A  B  D  E.  A  rod  m  n  moves  in  guides, 
which  only  permit  it  to  ascend  and  descend  perpendicularly. 
Its  extremity  m  rests  upon  a  path  or  groove  raised  from  the 
face  of  the  wheel,  and  shaped  into  such  a  curve  that,  as  the 
wheel  revolves,  the  rod  mn  shall  be  moved  alternately  in  op- 
posite directions  through  the  guides,  with  the  required  veloci- 
ty. The  manner  in  which  the  velocity  varies  will  depend  on 
the  form  given  to  the  groove  or  channel  raised  upon  the  face 
of  the  wheel,  and  this  may  be  shaped  so  as  to  give  any  varia- 
tion to  the  motion  of  the  rod  m  n  which  may  be  required  for 
the  purpose  to  which  it  is  to  be  applied. 


CHAP.  XVIII.  MODIFICATION    OF    MOTION.  211 

The  rose-engine  in  the  turning  lathe  is  constructed  on  this 
principle.  It  is  also  used  in  spinning  machinery. 

It  is  often  necessary  that  the  rod  to  which  reciprocating 
motion  is  communicated,  shall  be  urged  by  the  same  force 
in  both  directions.  A  wheel  partially  furnished  with  teeth, 
acting  on  two  racks  placed  on  different  sides  of  it,  and  both 
connected  with  the  bar  or  rod  to  which  the  reciprocating  mo- 
tion1 is  to  be  communicated,  will  accomplish  this.  Such  an 
apparatus  is  represented  in  jig.  163.,  and  needs  no  further  ex- 
planation. 

Another  contrivance  for  the  same  purpose  is  shown  in  Jig. 
164.,  where  A  is  a  wheel  turned  by  a  winch  H,  and  connect- 
ed with  a  rod  or  beam  moving  in  guides  by  the  joint  a  b. 
As  the  wheel  A  is  turned  by  the  winch  II,  the  beam  is  moved 
between  the  guides  alternately  in  opposite  directions,  the  ex- 
tent of  its  range  being  governed  by  the  length  of  the  diame- 
ter of  the  wheel.  Such  an  apparatus  is  used  for  grinding 
and  polishing  plane  surfaces,  and  also  occurs  in  silk  ma- 
chinery. 

An  apparatus  applied  by  M.  Zureda  in  a  machine  for  prick- 
ing holes  in  leather  is  represented  \njig.  165.  The  wheel  A 
B  has  its  circumference  formed  into  teeth,  the  shape  of  which 
may  be  varied  according  to  the  circumstances  under  which  it 
is  to  be  applied.  One  extremity  of  the  rod  a  b  rests  upon  the 
teeth  of  the  wheel,  upon  which  it  is  pressed  by  a  spring  at  the 
other  extremity.  When  the  wheel  revolves,  it  communicates 
to  this  rod  a  reciprocating  rectilinear  motion. 

Leupold  has  applied  this  mechanism  to  move  the  pistons  of 
pumps.*  Upon  the  vertical  axis  of  a  horizontal  hydraulic 
wheel  is  fixed  another  horizontal  wheel,  which  is  furnished 
with  seven  teeth,  in  the  manner  of  a  crown-wheel.  (263.) 
These  teeth  are  shaped  like  inclined  planes,  the  intervals  be- 
tween them  being  equal  to  the  length  of  the  planes.  Pro- 
jecting arms  attached  to  the  piston-rods  rest  upon  the  crown 
of  this  wheel ;  and,  as  it  revolves,  the  inclined  surfaces  of  the 
teeth,  being  forced  under  the  arm,  raise  the  rod  upon  the  prin- 
ciple of  the  wedge.  To  diminish  the  obstruction  arising  from 
friction,  the  projecting  arms  of  the  piston-rods  are  provided 
with  rollers,  which  run  upon  the  teeth  of  the  wheel.  In  one 
revolution  of  the  wheel  each  piston  makes  as  many  ascents 
and  descents  as  there  are  teeth. 

*  Theatrum  Machinarum,  torn.  ii.  pi.  36.  fig. 


212  THE    ELEMENTS    OF   MECHANICS.        CHAP.  XVIII. 

(317.)  Wheel- work  furnishes  numerous  examples  of  con- 
tinued circular  motion  round  one  axis,  producing  continued 
circular  motion  round  another.  If  the  axles  be  in  parallel 
directions,  and  not  too  distant,  rotation  may  be  transmitted 
from  one  to  the  other  by  two  spur-wheels  (263.);  and  the  rela- 
tive velocities  may  be  determined  by  giving  a  corresponding 
proportion  to  the  diameter  of  the  wheels. 

If  a  rotatory  motion  is  to  be  communicated  from  one  axis 
to  another  parallel  to  it,  and  at  any  considerable  distance, 
it  cannot  in  practice  be  accomplished  by  wheels  alone,  for 
their  diameters  would  be  too  large.  In  this  case,  a  strap  or 
chain  is  carried  round  the  circumferences  of  both  wheels.  If 
they  are  intended  to  turn  in  the  same  direction,  -the  strap  is 
arranged  as  in  Jig,  100. ;  but  if  in  contrary  directions,  it  is 
.crossed  as  in  Jig.  101.  In  this  case,  as  with  toothed  wheels, 
the  relative  velocities  are  determined  by  the  proportion  of  the 
diameters  of  the  wheels. 

If  the  axles  be  distant  and  not  parallel,  the  cord,  by  which 
the  motion  is  transmitted,  must  be  passed  over  grooved 
wheels,  or  fixed  pulleys,  properly  placed  between  the  two 
axles. 

It  may  happen  that  the  strain  upon  the  wheel,  to  which  the 
motion  is  to  be  transmitted,  is  too  great  to  allow  of  a  strap  or 
cord  being  used.  In  this  case,  a  shaft  extending  from  the  one 
axis  to  another,  and  carrying  two  bevelled  wheels  (263.),  will 
accomplish  the  object.  One  of  these  bevelled  wheels  is 
placed  upon  the  shaft  near  to,  and  in  connection  with,  the 
wheel  from  which  the  motion  is  to  be  taken,  and  the  other  at 
a  part  of  it  near  to,  and  in  connection  with,  that  wheel  to  which 
the  motion  is  to  be  conveyed,  Jig.  166. 

The  methods  of  transmitting  rotation  from  one  axis  to 
another  perpendicular  to  it,  by  crown  and  by  bevelled  wheels, 
have  been  explained  in  (263.) 

The  endless  screw  (299.)  is  a  machine  by  which  a  rotatory 
motion  round  one  axis  may  communicate  a  rotatory  motion 
round  another  perpendicular  to  it.  The  power  revolves 
round  an  axis  coinciding  with  the  length  of  the  screw,  and 
the  axis  of  the  wheel  driven  by  the  screw  is  at  right  angles 
to  this. 

The  axis  to  which  rotation  is  to  be  given,  or  from  which  it 
is  to  be  taken,  is  sometimes  variable  in  its  position.  In  such 
cases,  an  ingenious  contrivance,  called  a  universal  joint,  in- 
vented by  the  celebrated  Dr.  Hooke,  may  be  used.  The  two 


C1IAF.    XVIII.  UNIVERSAL   JOINT.  213 

shafts  or  axles  A  ^,fg.  167.,  between  which  the  motion  is  to 
be  communicated,  terminate  in  semicircles,  the  diameters  of 
which,  C  D  and  E  F,  are  fixed  in  the  form  of  a  cross,  their 
extremities  moving  freely  in  bushes  placed  in  the  extremities 
of  the  semicircles.  Thus,  while  the  central  cross  remains 
unmoved,  the  shaft  A  and  its  semicircular  end  may  revolve 
round  C  D  as  an  axis ;  and  the  shaft  B  and  its  semicircular 
end  may  revolve  round  E  F  as  an  axis.  If  the  shaft  A  be 
made  to  revolve  without  changing  its  direction,  the  points 
C  D  will  move  in  a  circle  whose  centre  is  at  the  middle  of  the 
cross.  The  motion  thus  given  to  the  cross  will  cause  the 
points  E  F  to  move  in  another  circle  round  the  same  centre, 
and  hence  the  shaft  B  will  be  made  to  revolve. 

This  instrument  will  not  transmit  the  motion  if  the  angle 
under  the  directions  of  the  shafts  be  less  than  140°.  In  this 
case  a  double  joint,  as  represented  in  Jig.  168.,  will  answer 
the  purpose.  This  consists  of  four  semicircles  united  by  two 
crosses,  and  its  principle  and  operation  are  the  same  as  in  the 
last  case. 

Universal  joints  are  of  great  use  in  adjusting  the  position 
of  large  telescopes,  where,  while  the  observer  continues  to  look 
through  the  tube,  it  is  necessary  to  turn  endless  screws  or 
wheels,  whose  axes  are  not  in  an  accessible  position. 

The  cross  is  not  indispensably  necessary  in  the  universal 
joint.  A  hoop,  with  four  pins  projecting  from  it  at  four  points 
equally  distant  from  each  other,  or  dividing  the  circle  of  the 
hoop  into  four  equal  arches,  will  answer  the  purpose.  These 
pins  play  in  the  bushes  of  the  semicircles  in  the  same  manner 
as  those  of  the  cross. 

The  universal  joint  is  much  used  in  cotton-mills,  where 
shafts  are  carried  to  a  considerable  distance  from  the  prime 
mover,  and  great  advantage  is  gained  by  dividing  them  into 
convenient  lengths,  connected  by  a  joint  of  this  kind. 

(318.)  In  the  practical  application  of  machinery,  it  is  often 
necessary  to  connect  a  part  having  a  continued  circular  mo- 
tion with  another  which  has  a  reciprocating  or  alternate 
motion,  so  that  either  may  move  the  other.  There  are  many 
contrivances  by  which  this  may  be  effected. 

One  of  the  most  remarkable  examples  of  it  is  presented 
in  the  scapements  of  watches  and  clocks.  In  this  case, 
however,  it  can  scarcely  be  said  with  strict  propriety  that  it 
is  the  rotation  of  the  scapement-wheel  (266.)  which  commu- 
nicates the  vibration  to  the  balance-wheel  or  r>endulurrn. 


214  THE    ELEMENTS    Of    MECHANICS.       CHAP.    XVIII. 

That  vibration  is  produced  in  the  one  case  by  the  peculiar 
nature  of  the  spiral  spring  fixed  upon  the  axis  of  the  balance- 
wheel,  and,  in  the  other  case,  by  the  gravity  of  the  pendulum. 
The  force  of  the  scapement*wheel  only  maintains  the  vibra- 
tion, and  prevents  its  decay  by  friction  and  atmospheric 
resistance,  Nevertheless,  between  the  two  parts  thus  moving, 
there  exists  a  mechanical  connection,  which  is  generally 
brought  within  the  class  of  contrivances  now  under  consid- 
eration. 

A  beam  vibrating  on  an  axis,  and  driven  by  the  piston  of  a 
steam-engine,  or  any  other  power,  may  communicate  rotatory 
motion  to  an  axis,  by  a  connector  and  a  crank.  This  appara- 
tus has  been  already  described  in  (311.)  Every  steam-engine 
which  works  by  a  beam  affords  an  example  of  this.  The 
working  beam  is  generally  placed  over  the  engine,  the  piston 
rod  being  attached  to  one  end  of  it,  while  the  connecting 
rod  unites  the  other  end  with  the  crank.  In  boat-engines, 
however,  this  position  would  be  inconvenient,  requiring  more 
room  than  could  easily  be  spared.  The  piston  rod,  in  these 
cases,  is,  therefore,  connected  with  the  end  of  the  beam  by 
long  rods,  and  the  beam  is  placed  beside  and  below  the 
engine.  Tne  use  of  a  fly-wheel  here  would  also  be  objection- 
able. The  effect  of  the  dead  points  explained  in  (311.)  is 
avoided  without  the  aid  of  a  fly,  by  placing  two  cranks  upon 
the  revolving  axle,  and  working  them  by  two  pistons.  The 
cranks  are  so  placed  that  when  either  is  at  its  dead  point, 
the  other  is  in  its  most  favorable  position. 

A  wheel  A,  Jig.  169.,  armed  with  wipers,  acting  upon  a 
sledge-hammer  B,  fixed  upon  a  centre  or  axle  C,  will,  by  a 
continued  rotatory  motion,  give  the  hammer  the  reciprocating 
motion  necessary  for  the  purposes  to  which  it  is  applied. 
The  manner  in  which  this  acts  must  be  evident  on  inspecting 
the  figure. 

The  treddle  of  the  lathe  furnishes  an  obvious  example  of 
a  vibrating  circular  motion  producing  a  continued  circular 
one.  The  treddle  acts  upon  a  crank,  which  gives  motion  to 
the  principal  wheel,  in  the  same  manner  as  already  described 
in  reference  to  the  working  beam  and  crank  in  the  steam- 
engine. 

By  the  following  ingenious  mechanism,  an  alternate  or 
vibrating  force  may  be  made  to  communicate  a  circular  mo- 
tion continually  in  the  same  direction.  Let  A  B,fig-  170., 
be  an  axis  receiving  an  alternate  motion  from  some  force 


CHAP.  XVIII.  MODIFICATION    OP    MOTION.  215 

applied  to  it,  such  as  a  swinging  weight.  Two  ratchet 
wheels  (253.)  m  and  n  are  fixed  on  this  axle,  their  teeth 
being  inclined  in  opposite  directions.  Two  toothed  wheels  C 
and  D  are  likewise  placed  upon  it,  but  so  arranged  that  they 
turn  upon  the  axle  with  a  little  friction.  These  wheels  car- 
ry two  catches  jp,  q,  which  fall  into  the  teeth  of  the  ratchet 
wheels  m,  n,  but  fall  on  opposite  sides,  conformably  to  the  in- 
clination of  the  teeth  already  mentioned.  The  effect  of  these 
catches  is,  that  if  the  axis  be  made  to  revolve  in  one  direction 
one  of  the  two  toothed  wheels  is  always  compelled  (by  the 
catch  against  which  the  motion  is  directed)  to  revolve  with  it, 
while  the  other  is  permitted  to  remain  stationary  in  obedience 
to  any  force  sufficiently  great  to  overcome  its  friction  with 
the  axle  on  which  it  is  placed.  The  wheels  C  and  D  are 
both  engaged  by  bevelled  teeth  (263.)  with  the  wheel  E. 

According  to  this  arrangement,  in  whichever  direction  the 
axis  A  B  is  made  to  revolve,  the  wheel  E  will  continually 
turn  in  the  same  direction,  and,  therefore,  if  the  axle  A  B 
be  made  to  turn  alternately  in  the  one  direction  and  the 
other,  the  wheel  E  will  not  change  the  direction  of  its  motion. 
Let  us  suppose  the  axle  A  B  is  turned  against  the  catch  p, 
The  wheel  C  will  then  be  made  to  turn  with  the  axle.  This 
will  drive  the  wheel  E  in  the  same  direction.  The  teeth  on 
the  opposite  side  of  the  wheel  E  being  engaged  with  those 
of  the  wheel  D,  the  latter  will  be  turned  upon  the  axle,  the 
friction,  which  alone  resists  its  motion  in  that  direction,  being 
overcome.  Let  the  motion  of  the  axle  A  B  be  now  reversed. 
Since  the  teeth  of  the  ratchet  wheel  n  are  moved  against  the 
catch  q,  the  wheel  D  will  be  compelled  to  revolve  with 
the  axle.  The  wheel  E  will  be  driven  in  the  same  direction 
as  before,  and  the  wheel  C  will  be  moved  on  the  axle  A  B, 
and  in  a  contrary  direction  to  the  motion  of  the  axle,  the 
friction  being  overcome  by  the  force  of  the  wheel  E,  Thus, 
while  the  axle  A  B  is  turned  alternately  in  the  one  direction 
and  the  other,  the  wheel  E  is  constantly  moved  in  the  same 
direction. 

It  is  evident  that  the  direction  in  which  the  wheel  E  moves 
may  be  reversed  by  changing  the  position  of  the  ratchet 
wheels  and  catches. 

(319.)  It  is  often  necessary  to  communicate  an  alternate 
circular  motion,  like  that  of  a  pendulum,  by  means  of  an 
alternate  motion  in  a  straight  line.  A  remarkable  instance 
of  this  occurs  in  the  steam-engine,  The  moving  force  in 


216  THE    ELEMENTS    OF    MECHANICS.         CHAP.  XVIII. 

this  machine  is  the  pressure  of  steam,  which  impels  a  piston 
from  end  to  end  alternately  in  a  cylinder.  The  force  of  this 
piston  is  communicated  to  the  working  beam  by  a  strong  rod, 
which  passes  through  a  collar  in  one  end  of  the  piston. 
Since  it  is  necessary  that  the  steam  included  in  the  cylinder 
should  not  escape  between  the  piston  rod  and  the  collar 
through  which  it  moves,  and  yet  that  it  should  move  as  freely, 
and  be  subject  to  as  little  resistance,  as  possible,  the  rod  is 
turned  so  as  to  be  truly  cylindrical,  and  is  well  polished. 
It  is  evident  that,  under  these  circumstances,  it  must  not  be 
subject  to  any  lateral  or  cross  strain,  which  would  bend  it 
towards  one  side  or  the  other  of  the  cylinder.  But  the  end 
of  the  beam  to  which  it  communicates  motion,  if  connected 
immediately  with  the  rod  by  a  joint,  would  draw  it  alternately 
to  the  one  side  and  the  other,  since  it  moves  in  the  arc  of  a 
circle,  the  centre  of  which  is  at  the  centre  of  the  beam.  It 
is  necessary,  therefore,  to  contrive  some  method  of  connecting 
the  rod  and  the  end  of  the  beam,  so  that  while  the  one  shall 
ascend  and  descend  in  a  straight  line,  the  other  may  move 
in  the  circular  arc. 

The  method  which  first  suggests  itself  to  accomplish  this 
is,  to  construct  an  arch-head  upon  the  end  of  the  beam,  as  in 
Jig.  171.  Let  C  be  the  centre  on  which  the  beam  works, 
and  let  B  D  be  an  arch  attached  to  the  end  of  the  beam, 
being  a  part  of  a  circle  having  C  for  its  centre.  To  the 
highest  point  B  of  the  arch  a  chain  is  attached,  which  is 
carried  upon  the  face  of  the  arch  B  A,  and  the  other  end  of 
which  is  attached  to  the  piston  rod.  Under  these  circum- 
stances, it  is  evident  that  when  the  force  of  the  steam  impels 
the  piston  downwards,  the  chain  P  A  B  will  draw  the  end 
of  the  beam  down,  and  will,  therefore,  elevate  the  other  end. 

When  the  steam-engine  is  used  for  certain  purposes,  such 
as  pumping,  this  arrangement  is  sufficient.  The  piston  in 
that  case  is  not  forced  upwards  by  the  pressure  of  steam. 
During  its  ascent  it  is  not  subject  to  the  action  of  any  force 
of  steam,  and  the  other  end  of  the  beam  falls  by  the  weight 
of  the  pump-rods  drawing  the  piston,  at  the  opposite  end  A, 
to  the  top  of  the  cylinder.  Thus  the  machine  is  in  fact  pas- 
sive during  the  ascent  of  the  piston,  and  exerts  its  power 
only  during  the  descent. 

If  the  machine,  however,  be  applied  to  purposes  in  which 
a  constant  action  of  the  moving  force  is  necessary,  as  is  al- 
ways the  case  in  manufactures,  the  force  of  the  piston  must 


OHAP.    XVIII.     ALTERNATE    MOTION    OF    A    PISTON.  217 

drive  the  beam  in  its  ascent  as  well  as  in  its  descent.  The 
arrangement  just  described  cannot  effect  this ;  for  although 
a  chain  is  capable  of  transmitting  any  force,  by  which  its 
extremities  are  drawn  in  opposite  directions,  yet  it  is,  from 
its  flexibility,  incapable  of  communicating  a  force  which 
drives  one  extremity  of  it  towards  the  other.  In  the  one 
case,  the  piston  first  pulls  down  the  beam,  and  then  the  beam 
pulls  up  the  piston.  The  chain,  because  it  is  inextensible,  is 
perfectly  capable  of  both  these  actions ;  and,  being  flexible, 
it  applies  itself  to  the  arch-head  of  the  beam,  so  as  to  main- 
tain the  direction  of  its  force  upon  the  piston  continually  in 
the  same  straight  line.  But  when  the  piston  acts  upon  the 
beam  in  both  ways,  in  pulling  it  down  and  pushing  it  up,  the 
chain  becomes  inefficient,  being  from  its  flexibility  incapable 
of  the  latter  action. 

The  problem  might  be  solved  by  extending  the  length  of 
the  piston  rod,  so  that  its  extremity  shall  be  above  the  beam, 
and  using  two  chains ;  one  connecting  the  highest  point  of 
the  rod  with  the  lowest  point  of  the  arch-head,  and  the  other 
connecting  the  highest  point  of  the  arch-head  with  a  point 
on  the  rod  below  the  point  which  meets  the  arch-head  when 
the  piston  is  at  the  top  of  the  cylinder,^-.  172. 

The  connection  required  may  also  be  made  by  arming  the 
arch-head  with  teeth,  Jig.  173.,  and  causing  the  piston  rod 
to  terminate  in  a  rack.  In  cases  where,  as  in  the  steam-en- 
gine, smoothness  of  motion  is  essential,  this  method  is  objec- 
tionable ;  and  under  any  circumstances,  such  an  apparatus  is 
liable  to  rapid  wear. 

The  method  contrived  by  Watt,  for  connecting  the  motion 
of  the  piston  with  that  of  the  beam,  is  one  of  the  most  in- 
genious and  elegant  solutions  ever  proposed  for  a  mechanical 
problem.  He  conceived  the  motion  of  two  straight  rods 
A  B,  C  D,^g\  174.,  moving  on  centres  or  pivots  A  and  C, 
so  that  the  extremities  B  and  D  would  move  in  the  arcs  of 
circles,  having  their  centres  at  A  and  C.  The  extremities 
B  and  D  of  these  rods  he  conceived  to  be  connected  with  a 
third  rod  B  D  united  with  them  by  pivots  on  which  it  could 
turn  freely.  To  the  system  of  rods  thus  connected  let  an 
alternate  motion  on  the  centres  A  and  C  be  communicated  ; 
the  points  B  and  D  will  move  upwards  and  downwards  in 
the  arcs  expressed  by  the  dotted  lines,  but  the  middle  point 
P  of  the  connecting  rod  B  D  will  move  upwards  and  down- 
wards without  any  sensible  deviation  from  a  straight  line. 
19 


THE    ELEMENTS    OF    MECHANICS.        CHAP.  XVIII. 

To  prove  this  demonstratively  would  require  some  abstruse 
mathematical  investigation.  It  may,  however,  be  rendered 
in  some  degree  apparent  by  reasoning  of  a  looser  and  more 
popular  nature.  As  the  point  B  is  raised  to  E,  it  is  also  drawn 
aside  towards  the  right.  At  the  same  time,  the  other  extremity 
D  of  the  rod  B  D  is  raised  to  E7,  and  is  drawn  aside  towards 
the  left.  The  ends  of  the  rod  B  D  being  thus  at  the  same 
time  drawn  equally  towards  opposite  sides,  its  middle  point  P 
will  suffer  no  lateral  derangement,  and  will  move  directly 
upwards.  On  tho  other  hand,  if  B  be  moved  downwards  to 
F,  it  will  be  drawn  laterally  to  the  right,  while  D,  being 
moved  to  F7,  will  be  drawn,  to  the  left.  Hence,  as  before, 
the  middle  point  P  sustains  no  lateral  derangement,  but 
merely  descends.  Thus  as  the  extremities  B  and  D  move 
upwards  and  downwards  in  circles,  the  middle  point  P  moves 
upwards  and  downwards  in  a  straight  line.* 

The  application  of  this  geometrical  principle  in  the  steam- 
engine  evinces  much  ingenuity.  The  same  arm  of  the  beam 
usually  works  two  pistons,  that  of  the  cylinder  and  that  of 
the  air-pump.  The  apparatus  is  represented  on  the  arm  of 
the  beam  in  Jig.  175.  The  beam  moves  alternately  upwards 
and  downwards  on  its  axis  A.  Every  point  of  it,  therefore, 
describes  a  part  of  a  circle  of  which  A  is  the  centre.  Let 
B  be  the  point  which  divides  the  arm  A  G  into  two  equal 
parts  A  B  and  B  G  ;  and  let  C  D  be  a  straight  rod,  equal  in 
length  to  G  B,  and  fixed  on  a  centre  or  pivot  C  on  which  it 
is  at  liberty  to  play.  The  end  D  of  this  rod  is  connected  by 
a  straight  bar  with  the  point  B,  by  pivots  on  which  the  rod 
B  D  turns  freely.  If  the  beam  be  now  supposed  to  rise  and 
fall  alternately,  the  points  B  and  D  will  move  upwards  and 
downwards  in  circular  arcs,  and,  as  already  explained  with 
respect  to  the  points  B  D,  Jig.  174.,  the  middle  point  P  of 
the  connecting  rod  B  D  will  move  upwards  and  downwards 
without  lateral  deflection.  To  this  point  one  of  the  piston 
rods  which  are  to  be  worked  is  attached. 

To  comprehend  the  method  of  working  the  other  piston, 
conceive  a  rod  G  P7,  equal  in  length  to  B  D,  to  be  attached 
to  the  end  G  of  the  beam  by  a  pivot  on  which  it  moves  freely ; 

*  In  a  strictly  mathematical  sense,  the  path  of  the  point  P  is  a  curve,  and 
not  a  straight  line ;  but  in  the  play  given  to  it  in  its  application  to  the  steam- 
engine,  it  moves  through  a  part  only  of  its  entire  locus,  and  this  part  extend- 
ing equally  on  each  side  of  a  point  of  inflection,  the  radius  of  curvature  is 
infinite,  so  that  in  practice  the  deviation  from  a  straight  line,  when  proper  pro 
portions  are  observed  in  the  rods,  is  imperceptible 


CHAP.  XIX.  FRICTION.  219 

arid  let  its  extremity  P'  be  connected  with  D  by  another  rod 
P'  D,  equal  in  length  to  G  B,  and  playing  on  points  at  P'  and 
D.  The  piston  rod  of  the  cylinder  is  attached  to  the  point 
P7,  and  this  point  has  a  motion  precisely  similar  to  that  of  P, 
without  any  lateral  derangement,  but  with  a  range  in  the  per- 
pendicular direction  twice  as  great.  This  will  be  apparent 
by  conceiving  a  straight  line  drawn  from  the  centre  A  of  the 
beam  to  P',  which  will  also  pass  through  P.  Since  G  P'  is 
always  parallel  to  B  P,  it  is  evident  that  the  triangle  P'  G  A 
is  always  similar  to  P  B  A,  and  has  its  sides  and  angles  simi- 
larly placed,  but  those  sides  are  each  twice  the  magnitude  of 
the  corresponding  sides  of  the  other  triangle.  Hence  the 
point  P'  must  be  subject  to  the  same  changes  of  position  as 
the  point  P,  with  this  difference  only,  that  in  the  same  time 
it  moves  over  a  spaee  of  twice  the  magnitude.  In  fact,  the 
line  traced  by  P'  is  the  same  as  that  traced  by  P,  but  on  a 
scale  twice  as  large.  This  contrivance  is  usually  called  the 
parallel  motion,  but  the  same  name  is  generally  applied  to  all 
contrivances  by  which  a  circular  motion  is  made  to  produce 
-  ^ctilinear  one. 


CHAPTER   XIX. 

OF    FRICTION    AND    THE    RIGIDITY    OF    CORDAGE. 

(320.)  WITH  a  view  to  the  simplification  of  the  elementary 
theory  of  machines,  the  consideration  of  several  mechanical 
effects  of  great  practical  importance  has  been  postponed,  and 
the  attention  of  the  student  has  been  directed  exclusively  to 
the  way  in  which  the  moving  power  is  modified  in  being 
transmitted  to  the  resistance  independently  of  such  effects. 
A  machine  has  been  regarded  as  an  instrument  by  which  a 
moving  principle,  inapplicable  in  its  existing  state  to  the  pur- 
pose for  which  it  is  required,  may  be  changed  either  in  its 
velocity  or  direction,  or  in  some  other  character,  so  as  to  be 
adapted  to  that  purpose.  But  in  accomplishing  this,  the  sev- 
eral parts  of  the  machine  have  been  considered  as  possessing 
in  a  perfect  degree  qualities  which  they  enjoy  only  in  an  im- 
perfect degree  ;  and  accordingly  the  conclusions  to  which  by 
such  reasoning  we  are  conducted  are  infected  with  errors, 


220  THE  ELEMENTS  OF  MECHANICS.  CHAP.  XIX 

the  amount  of  which  will  depend  on  the  degree  in  which  the 
machinery  falls  short  of  perfection  in  those  qualities  which 
theoretically  are  imputed  to  it. 

Of  the  several  parts  of  a  machine,  some  are  designed  to 
move,  while  others  are  fixed  ;  and  of  those  which  move,  some 
have  motions  differing  in  quantity  and  direction  from  those 
of  others.  The  several  parts,  whether  fixed  or  movable,  are 
subject  to  various  strains  and  pressures,  which  they  are  in- 
tended to  resist.  These  forces  not  only  vary  according  to 
the  load  which  the  machine  has  to  overcome,  but  also  accord- 
ing to  the  peculiar  form  and  structure  of  the  machine  itself. 
During  the  operation,  the  surfaces  of  the  movable  parts  move 
in  immediate  contact  with  the  surfaces  either  of  fixed  parts 
or  of  parts  having  other  motions.  If  these  surfaces  were 
endued  with  perfect  smoothness  or  polish,  and  the  several 
parts  subject  to  strains  possessed  perfect  inflexibility  and  in- 
finite strength,  then  the  effects  of  machinery  might  be  practi- 
cally investigated  by  the  principles  already  explained.  But 
the  materials  of  which  every  machine  is  formed  are  endued 
with  limited  strength,  and  therefore  the  load  which  is  placed 
upon  it  must  be  restricted  accordingly,  else  it  will  be  liable 
to  be  distorted  by  the  flexure,  or  even  to  be  destroyed  by  the 
fracture  of  those  parts  which  are  submitted  to  an  undue 
strain.  The  surfaces  of  the  movable  parts,  and  those  surfaces 
with  which  they  move  in  contact,  cannot  in  practice  be  ren- 
dered so  smooth  but  that  such  roughness  and  inequality  will 
remain  as  sensibly  to  impede  the  motion.  To  overcome  such 
an  impediment  requires  no  inconsiderable  part  of  the  moving 
power.  This  part  is,  therefore,  intercepted  before  its  arrival 
at  the  working  point,  and  the  resistance  to  be  finally  overcome 
is  deprived  of  it.  The  property  thus  depending  on  the  im- 
perfect smoothness  of  surfaces,  and  impeding  the  motion  of 
bodies  whose  surfaces  are  in  immediate  contact,  is  called 
friction.  Before  we  can  form  a  just  estimate  of  the  effects 
of  machinery,  it  is*  necessary  to  determine  the  force  lost  by 
this  impediment,  and  the  laws  which,  under  different  circum- 
stances, regulate  that  loss. 

When  cordage  is  engaged  in  the  formation  of  any  part  of 
a  machine,  it  has  hitherto  been  considered  as  possessing  per- 
fect flexibility.  This  is  not  the  case  in  practice ;  and  the 
want  of  perfect  flexibility,  which  is  called  rigidity,  renders  a 
certain  quantity  of  force  necessary  to  bend  a  cord  or  rope 
over  the  surface  of  an  axle  or  the  groove  of  a  wheel.  During 


CHAP.    XIX.  FRICTION.  221 

the  motion  of  the  tope,  a  different  part  of  it  must  thus  be  con- 
tinually bent,  and  the  force  which  is  expended  in  producing 
the  necessary  flexure  must  be  derived  from  the  moving  power, 
and  is  thus  intercepted  on  its  way  to  the  working  point.  In 
calculating  the  effects  of  cordage,  due  regard  must  be  had  to 
this  waste  of  power ;  and  therefore  it  is  necessary  to  inquire 
into  the  laws  which  govern  the  flexure  of  imperfectly  flexible 
ropes,  and  the  way  in  which  these  affect  the  machines  in 
which  ropes  are  commonly  used. 

To  complete,  therefore,  the  elementary  theory  of  machine- 
ry, we  propose  in  the  present  and  following  chapter  to  explain 
the  principal  laws  which  determine  the  effects  of  friction,  the 
rigidity  of  cordage,  and  the  strength  of  materials. 

(321.)  If  a  horizontal  plane  surface  were  perfectly  smooth, 
and  free  from  the  smallest  inequalities,  and  a  body  having  a 
flat  surface,  also  perfectly  smooth,  were  placed  upon  it,  any 
force  applied  to  the  latter  would  put  it  in  motion,  and  that 
motion  would  continue  undimjnished  as  long  as  the  body 
would  remain  upon  the  smooth  horizontal  surface.  But  if 
this  surface,  instead  of  being  every  where  perfectly  even,  had 
in  particular  places  small  projecting  eminences,  a  certain 
quantity  of  force  would  be  necessary  to  carry  the  moving  body 
over  these,  and  a  proportional  diminution  in  its  rate  of  motion 
would  ensue.  Thus,  if  such  eminences  were  of  frequent 
occurrence,  each  would  deprive  the  body  of  apart  of  its  speed, 
so  that  between  that  and  the  next  it  would  move  with  a  less 
velocity  than  it  had  between  the  same  and  the  preceding  one. 
This  decrease  being  continued  by  a  sufficient  number  of  such 
eminences  encountering  the  body  in  succession,  the  velocity 
would  at  last  be  so  much  diminished,  that  the  body  would  not 
have  sufficient  force  to  carry  it  over  the  next  eminence,  and 
its  motion  would  thus  altogether  cease. 

Now,  instead  of  the  eminences  being  at  a  considerable  dis- 
tance asunder,  suppose  them  to  be  contiguous,  and  to  be 
spread  in  every  direction  over  the  horizontal  plane,  and  also 
suppose  corresponding  eminences  to  be  upon  the  surface  of  the 
moving  body ;  these  projections  incessantly  encountering  one 
another  will  continually  obstruct  the  motion  of  the  body,  and 
will  gradually  diminish  its  velocity,  until  it  be  reduced  to  a 
state  of  rest. 

Such  is  the  cause  of  friction.  The  amount  of  this  resist- 
ing force  increases  with  the  magnitude  of  these  asperities,  or 
with  the  roughness  of  the  surfaces  ;  but  it  does  not  solely  de- 
19* 


THE  ELEMENTS  OF  MECHANICS.      CHAP.  XIX. 

pend  on  this.  The  surfaces  remaining  the  same,  a  little  re- 
flection on  the  method  of  illustration  just  adopted,  will  show 
that  the  amount  of  friction  ought  also  to  depend  upon  the  force 
with  which  the  surfaces  moving  one  upon  the  other  are  press- 
ed together.  It  is  evident,  that  as  the  weight  of  the  body 
supposed  to  move  upon  the  horizontal  plane  is  increased,  a  pro- 
portionally greater  force  will  be  necessary  to  carry  it  over  the 
obstacles  which  it  encounters,  and  therefore  it  will  the  more 
speedily  be  deprived  of  its  velocity  and  reduced  to  a  state  of 
rest. 

(322.)  Thus  we  might  predict  with  probability,  that  which 
accurate  experimental  inquiry  proves  to  be  true,  that  the  re- 
sistance from  friction  depends  conjointly  on  the  roughness  of 
the  surfaces  and  the  force  of  the  pressure.  When  the  sur- 
faces are  the  same,  a  double  pressure  will  produce  a  double 
amount  of  friction,  a  treble  pressure  a  treble  amount  of  fric- 
tion, and  so  on. 

Experiment  also,  however,  gives  a  result  which,  at  least  at 
first  view,  might  not  have  been  anticipated  from  the  mode  of 
illustration  we  have  adopted.  It  is  found  that  the  resistance 
arising  from  friction  does  not  at  all  depend  on  the  magnitude 
of  the  surface  of  contact ;  but  provided  the  nature  of  the  sur- 
faces and  the  amount  of  pressure  remain  the  same,  this  resist- 
ance will  be  equal,  whether  the  surfaces  which  move  one  upon 
the  other  be  great  or  small.  Thus,  if  the  moving  body  be  a 
flat  block  of  wood,  the  face  of  which  is  equal  to  a  square  foot 
in  magnitude,  and  the  edge  of  which  does  not  exceed  a  square 
inch,  it  will  be  subject  to  the  same  amount  of  friction,  wheth- 
er it  move  upon  its  broad  face  or  upon  its  narrow  edge.  If 
we  consider  the  effect  of  the  pressure  in  each  case,  we  shall 
be  able  to  perceive  why  this  must  be  the  case.  Let  us  sup- 
pose the  weight  of  the  block  to  be  144  ounces.  When  it 
rests  upon  its  face,  a  pressure  to  this  amount  acts  upon  a  sur- 
face of  144  square  inches,  so  that  a  pressure  of  one  ounce 
acts  upon  each  square  inch.  The  total  resistance  arising  from 
friction  will,  therefore,  be  144  times  that  resistance  which 
would  be  produced  by  a  surface  of  one  square  inch  under  a 
pressure  of  one  ounce.  Now,  suppose  the  block  placed  upon 
its  edge,  there  is  then  a  pressure  of  144  ounces  upon  a  sur- 
face equal  to  one  square  inch.  But  it  has  been  already  shown, 
that  when  the  surface  is  the  same,  the  friction  must  increase 
in  proportion  to  the  pressure.  Hence  we  infer,  that  the  fric- 
tion produced  in  the  present  case  is  144  times  the  friction 


CHAP.    XIX.  FRICTION.  223 

which  would  be  produced  by  a  pressure  of  one  ounce  acting 
on  one  square  inch  of  surface,  which  is  the  same  resistance 
as  that  which  the  body  was  proved  to  be  subject  to  when  rest- 
ing on  its  face. 

These  two  laws,  that  friction  is  independent  of  the  magni- 
tude of  the  surface,  and  is  proportional  to  the  pressure,  when 
the  quality  of  the  surfaces  is  the  same,  are  useful  in  practice, 
and  generally  true.  In  very  extreme  cases  they  are,  however, 
in  error.  When  the  pressure  is  very  intense,  in  proportion  to 
the  surface,  the  friction  is  somewhat  less  than  it  would  be  by 
these  laws ;  and  when  it  is  very  small  in  proportion  to  the  sur- 
face, it  is  somewhat  greater. 

(323.)  There  are  two  methods  of  establishing  by  experi- 
ment the  laws  of  friction,  which  have  been  just  explained. 

First.  The  surfaces  between  which  the  friction  is  to  be 
determined  being  rendered  perfectly  flat,  let  one  be  fixed  in 
the  horizontal  position  on  a  table  T  T',  Jig.  176. ;  and  let 
the  other  be  attached  to  the  bottom  of  a  box  B  C,  adapted  to 
receive  weights,  so  as  to  vary  the  pressure.  Let  a  silken  cord 
S  P,  attached  to  the  box,  be  carried  parallel  to  the  table  over 
a  wheel  at  P,  and  let  a  dish  D  be  suspended  from  it.  If  no 
friction  existed  between  the  surfaces,  the  smallest  weight  ap- 
pended to  the  cord  would  draw  the  box  towards  P  with  a  con- 
tinually increasing  speed.  But  the  friction  which  always  ex- 
ists interrupts  this  effect,  and  a  small  weight  may  act  upon 
the  string  without  moving  the  box  at  all.  Let  weights  be 
put  in  the  dish  D,  until  a  sufficient  force  is  obtained  to  over- 
come the  friction  without  giving  the  box  an  accelerated  motion. 
Such  a  weight  is  equivalent  to  the  amount  of  the  friction. 

The  amount  of  the  weight  of  the  box  being  previously  as- 
certained, let  this  weight  be  now  doubled  by  placing  addition- 
al weights  in  the  box.  The  pressure  will  thus  be  doubled, 
and  it  will  be  found  that  the  weight  of  the  dish  D  and  its 
load,  which  before  was  able  to  overcome  the  friction,  is  now 
altogether  inadequate  to  it,  Let  additional  weights  be  placed 
in  the  dish,  until  the  friction  be  counteracted  as  before,  and  it 
will  be  observed,  that  the  whole  weight  necessary  to  produce 
this  effect  is  exactly  twice  the  weight  which  produced  it  in 
the  former  case.  Thus  it  appears  that  a  double  amount  of 
pressure  produces  a  double  amount  of  friction  ;  and  in  a  sim- 
ilar way  it  may  be  proved,  that  any  proposed  increase  or  de- 
crease of  the  pressure  will  be  attended  with  a  proportionate 
variation  in  the  amount  of  the  friction. 


224  TUB    ELEMENTS    OF   MECHANICS.          CHAP.    XIX 

Second.  Let  one  of  the  surfaces  be  attached  to  a  flat  plane 
A  B,Jig.  177.,  which  can  be  placed  at  any  inclination  with  an 
horizontal  plane  B  C,  the  other  surface  being,  as  before,  at- 
tached to  the  box  adapted  to  receive  weights.  The  box  be- 
ing placed  upon  the  plane,  let  the  latter  be  slightly  elevated. 
The  tendency  of  the  box  to  descend  upon  A  B  will  bear  the 
same  proportion  to  its  entire  weight  as  the  perpendicular  A  E 
bears  to  the  length  of  the  plane  A  B  (2HG.).  Thus  if  the 
length  A  B  be  30  inches,  and  the  height  A  E  be  three  inches, 
that  is,  a  twelfth  part  of  the  length,  then  the  tendency  of  the 
weight  to  move  down  the  plane  is  equal  to  a  twelfth  part  of 
its  whole  amount.  If  the  weight  were  twelve  ounces,  and 
the  surfaces  perfectly  smooth,  a  force  of  one  ounce  acting  up 
the  plane  would  be  necessary  to  prevent  the  descent  of  the 
weight. 

In  this  case  also,  the  pressure  on  the  plane  will  be  repre- 
sented by  the  length  of  the  base  B  E  (280.),  thai  is,  it  will 
bear  the  same  proportion  to  the  whole  weight  as  B  E  bears  to 
B  A.  The  relative  amounts  of  the  weight,  the  tendency  to 
descend,  and  the  pressure,  will  always  be  exhibited  by  the  rel- 
ative lengths  of  A  B,  A  E,  and  B  E. 

This  being  premised,  let  the  elevation  of  the  plane  A  B  be 
gradually  increased,  until  the  tendency  of  the  weight  to  de- 
scend just  overcomes  the  friction,  but  not  so  much  as  to  allow 
the  box  to  descend  with  accelerated  speed.  The  proportion 
of  the  whole  weight,  which  then  acts  down  the  plane,  will  be 
found  by  measuring  the  height  A  E,  and  the  pressure  will  be 
determined  by  measuring  the  base  B  E.  Now  let  the  weight 
in  the  box  be  increased,  and  it  will  be  found  that  the  same 
elevation  is  necessary  to  overcome  the  friction ;  nor  will  this 
elevation  suffer  any  change,  however  the  pressure  or  the 
magnitude  of  the  surfaces  which  move  in  contact  may  be 
varied. 

Since,  therefore,  in  all  these  cases,  the  height  A  E  and  the 
base  B  E  remain  the  same,  it  follows  that  the  proportion  be- 
tween the  friction  and  pressure  is  undisturbed. 

(324.)  The  law  that  friction  is  proportional  to  the  pressure, 
has  been  questioned  by  the  late  professor  Vince  of  Cambridge, 
who  deduced  from  a  series  of  experiments,  that  although  the 
friction  increases  with  the  pressure,  yet  that  it  increases  in  a 
somewhat  less  ratio ;  and  from  this  it  would  follow,  that  the 
variation  of  the  surface  of  contact  must  produce  some  effect 
upon  the  amount  of  friction.  The  law  as  we  have  explained 


CHAP.    XIX,  FRICTION.  225 


it,  however,  is  sufficiently  near  the  truth  for  most  practical 
purposes. 

(325.)  There  are  several  circumstances  regarding  the 
quality  of  the  surfaces,  which  produce  important  effects  on 
the  quantity  of  friction,  and  which  ought  to  be  noticed 
here. 

This  resistance  is  different  in  the  surfaces  of  different  sub- 
stances. When  the  surfaces  are  those  of  wood  newly  planed, 
it  amounts  to  about  half  the  pressure,  but  is  different  in  dif- 
ferent kinds  of  wood.  The  friction  of  metallic  surfaces  is 
about  one  fourth  of  the  pressure. 

In  general,  the  friction  between  the  surfaces  of  bodies  of 
different  kinds  is  less  than  between  those  of  the  same  kind. 
Thus,  between  wood  and  metal,  the  friction  is  about  one  fifth 
of  the  pressure. 

It  is  evident  that  the  smoother  the  surfaces  are,  the  less 
will  be  the  friction.  On  this  account,  the  friction  of  surfaces, 
when  first  V  'ought  into  contact,  is  often  greater  than  after 
their  attrition  has  been  continued  for  a  certain  time,  because 
that  process  has  a  tendency  to  remove  and  rub  off  those  mi- 
nute asperities  and  projections  on  which  the  friction  depends. 
But  this  has  a  limit,  and  after  a  certain  quantity  of  attrition, 
the  friction  ceases  to  decrease.  Newly  planed  surfaces  of 
wood  have  at  first  a  degree  of  friction  which  is  equal  to  half 
the  entire  pressure,  but  after  they  are  worn  by  attrition,  it  is 
reduced  to  a  third. 

If  the  surfaces  in  contact  be  placed  with  their  grains  in  the 
same  direction,  the  friction  will  be  greater  than  if  the  grains 
cross  each  other. 

Smearing  the  surfaces  with  unctuous  matter,  diminishes 
the  friction,  probably  by  filling  the  cavities  between  the  minute 
projections  which  produce  the  friction. 

When  the  surfaces  are  first  placed  in  contact,  the  friction  is 
less  than  when  they  are  suffered  to  rest  so  for  some  time ; 
this  is  proved  by  observing  the  force  which  in  each  case  is 
necessary  to  move  the  one  upon  the  other,  that  force  being 
less  if  applied  at  the  first  moment  of  contact  than  when  the 
contact  has  continued.  This,  however,  has  a  limit.  There 
is  a  certain  time,  different  in  different  substances,  within 
which  this  resistance  attains  its  greatest  amount.  In  surfaces 
of  wood,  this  takes  place  in  about  two  minutes  ;  in  metals,  the 
time  is  imperceptibly  short ;  and  when  a  surface  of  wood  is 
placed  upon  a  surface  of  metal,  it  continues  to  increase  for 


226  THE    ELEMENTS    OF    MECHANICS.  CHAP.  XIX. 

several  days.  The  limit  is  larger  when  the  surfaces  are  great, 
and  belong  to  substances  of  different  kinds. 

The  velocity  with  which  the  surfaces  move  upon  one  another 
produces  but  little  effect  upon  the  friction. 

(326.)  There  are  several  ways  in  which  bodies  may  move 
one  upon  the  other,  in  which  friction  will  produce  different 
effects.  The  principal  of  these  are,  first,  the  case  where  one 
body  slides  over  another;  the  second,  where  a  body  having  a 
round  form  rolls  upon  another ;  and,  thirdly,  where  an  axis 
revolves  within  a  hollow  cylinder,  or  the  hollow  cylinder  re- 
volves upon  the  axis. 

With  the  same  amount  of  pressure  and  a  like  quality  of 
surface,  the  quantity  of  friction  is  greatest  in  the  first  case 
and  least  in  the  second.  The  friction  in  the  second  case  also 
depends  on  the  diameter  of  the  body  which  rolls,  and  is  small 
in  proportion  as  that  diameter  is  great.  Thus  a  carriage  with 
large  wheels  is  less  impeded  by  the  friction  of  the  road  than 
one  with  small  wheels. 

In  the  third  case,  the  leverage  of  the  wheel  aids  the  power 
in  overcoming  the  friction.  "Let  Jig.  178.  represent  a  section 
of  the  wheel  and  axle ;  let  C  be  the  centre  of  the  axle,  and 
let  B  E  be  the  hollow  cylinder  in  the  nave  of  the  wheel  in 
which  the  axle  is  inserted.  If  B  be  the  part  on  which  the 
axle  presses,  and  the  wheel  turn  in  the  direction  N  D  M,  the 
friction  will  act  at  B  in  the  direction  B  F,  arid  with  the  lever- 
age B  C.  The  power  acts  against  this  at  D  in  the  direction 
D  A,  and  with  the  leverage  D  C.  It  is  therefore  evident, 
that  as  D  C  is  greater  than  B  C,  in  the  same  proportion 
does  the  power  act  with  mechanical  advantage  on  the  fric- 
tion. 

(327.)  Contrivances  for  diminishing  the  effects  of  friction 
depend  on  the  properties  just  explained,  the  motion  of  rolling 
being  as  much  as  possible  substituted  for  that  of  sliding;  and 
where  the  motion  of  rolling  cannot  be  applied,  that  of  a  wheel 
upon  its  axle  is  used.  In  some  cases,  both  these  motions  are 
combined. 

If  a  heavy  load  be  drawn  upon  a  plane  in  the  manner  of  a 
sledge,  the  motion  will  be  that  of  sliding,  the  species  which 
is  attended  with  the  greatest  quantity  of  friction ;  but  if  the 
load  be  placed  upon  cylindrical  rollers,  the  nature  of  the  mo- 
tion is  changed,  and  becomes  that  in  which  there  is  the  least 
quantity  of  friction.  Tims  large  blocks  of  stone,  or  heavy 
beams  of  timber,  which  would  require  an  enormous  power  to 


CHAP.  XIX.  FRICTION.  227 

move  them  on  a  level  road,  are  easily  advanced  when  rollers 
are  put  under  them. 

When  very  heavy  weights  are  to  be  moved  through  small 
spaces,  this  method  is  used  with  advantage ;  but  when  loads 
are  to  be  transported  to  considerable  distances,  the  process  is 
inconvenient  and  slow,  owing  to  the  necessity  of  continually 
replacing  the  rollers  in  front  of  the  load  as  they  are  left  be- 
hind by  its  progressive  advancement. 

The  wheels  of  carriages  may  be  regarded  as  rollers  which 
are  continually  carried  forward  with  the  load.  In  addition 
to  the  friction  of  the  rolling  motion  on  the  road,  they  have,  it 
is  true,  the  friction  of  the  axle  in  the  nave  ;  but,  on  the  other 
hand,  they  are  free  from  the  friction  of  the  rollers  with  the 
under  surface  of  the  load,  or  the  carriage  in  which  the  load 
is  transported.  The  advantage  of  wheel  carriages  in  dimin- 
ishing the  effects  of  friction,  is  sometimes  attributed  to  the 
slowness  with  which  the  axle  moves  within  the  box,  compar- 
ed with  the  rate  at  which  the  wheel  moves  over  the  road  ;  but 
this  is  erroneous.  The  quantity  of  friction  does  not  in  any 
case  vary  considerably  with  the  velocity  of  the  motion,  but 
least  of  all  does  it  in  that  particular  kind  of  motion  here  con- 
sidered. 

In  certain  cases,  where  it  is  of  great  importance  to  remove 
the  effects  of  friction,  a  contrivance  called  friction-wheels,  or 
friction-rollers,  is  used.  The  axle  of  a  friction-wheel,  instead 
of  revolving  within  a  hollow  cylinder,  which  is  fixed,  rests 
upon  the  edges  of  wheels  which  revolve  with  it ;  the  species 
of  motion  thus  becomes  that  in  which  the  friction  is  of  least 
amount. 

Let  A  B  and  D  C,  Jig.  179.,  be  two  wheels  revolving  on 
pivots  P  Q,  with  as  little  friction  as  possible,  and  so  placed 
that  the  axle  O  of  a  third  wheel  E  F  may  rest  between  their 
edges.  As  the  wheel  E  F  revolves,  the  axle  O,  instead  of 
grinding  its  surface  on  the  surface  on  which  it  presses,  car- 
ries that  surface  with  it,  causing  the  wheels  A  B,  C  D,  to  re- 
volve. 

In  wheel  carriages,  the  roughness  of  the  road  is  more 
easily  overcome  by  large  wheels  than  by  small  ones.  The 
cause  of  this  arises  partly  from  the  large  wheels  not  being  so 
liable  to  sink  into  holes  as  small  ones,  but  more  because,  in 
surmounting  obstacles,  the  load  is  elevated  less  abruptly.  This 
will  be  easily  understood  by  observing  the  curves  in  Jig.  180., 
which  represent  the  elevation  of  the  axle  in  each  case. 


228  THE  ELEMENTS  OF  MECHANICS.     CHAP.  XIX. 

(328.)  If  a  carriage  were  capable  of  moving  on  a  road 
without  friction,  the  most  advantageous  direction  in  which  a 
force  could  be  applied  to  draw  it  would  be  parallel  to  the 
road.  When  the  motion  is  impeded  by  friction,  it  is  better, 
however,  that  the  line  of  draught  should  be  inclined  to  the 
road,  so  that  the  drawing  force  may  be  expended  partly  in 
lessening  the  pressure  on  the  road,  and  partly  in  advancing 
the  load. 

Let  W,  Jig.  181.,  be  a  load  which  is  to  be  moved  upon  the 
plane  surface  A  B.  If  the  drawing  force  be  applied  in  the 
direction  C  D,  parallel  to  the  plane  A  B,  it  will  have  to  over- 
come the  friction  produced  by  the  pressure  of  the  whole 
weight  of  the  load  upon  the  plane;  but  if  it  be  inclined  up- 
wards in  the  direction  C  E,  it  will  be  equivalent  to  two  forces 
expressed  (74.)  by  C  G  and  C  F.  The  part  C  G  has  the 
effect  of  lightening  the  pressure  of  the  carriage  upon  the 
road,  and  therefore  of  diminishing  the  friction  in  the  same 
proportion.  The  part  C  F  draws  the  load  along  the  plane. 
Since  C  F  is  less  than  C  E  or  C  D,  the  whole  moving  force, 
it  is  evident  that  a  part  of  the  force  of  draught  is  lost  by  this 
obliquity  ;  but,  on  the  other  hand,  a  part  of  the  opposing  re- 
sistance is  also  removed.  If  the  latter  exceed  the  former,  an 
advantage  will  be  gained  by  the  obliquity  ;  but  if  the  former 
exceed  the  latter,  force  will  be  lost. 

By  mathematical  reasoning,  founded  on  these  considera- 
tions, it  is  proved  that  the  best  angle  of  draught  is  exactly 
that  obliquity  which  should  be  given  to  the  road  in  order  to 
enable  the  carriage  to  move  of  itself.  This  obliquity  is 
sometimes  called  the  angle  of  repose,  and  is  that  angle  which 
determines  the  proportion  of  the  friction  to  the  pressure  in 
the  second  method  explained  in  (323.)  The  more  rough 
the  road  is,  the  greater  will  this  angle  be ;  and  therefore  it 
follows,  that  on  bad  roads  the  obliquity  of  the  traces  to  the 
road  should  be  greater  than  on  good  ones.  On  a  smooth 
Macadamized  way,  a  very  slight  declivity  would  cause  a  car- 
riage to  roll  by  its  own  weight :  hence,  in  this  case,  the 
traces  should  be  nearly  parallel  to  the  road. 

In  rail-roads,  for  like  reasons,  the  line  of  draught  should 
be  parallel  to  the  road,  or  nearly  so. 

(329.)  When  ropes  or  cords  form  a  part  of  machinery,  the 
effects  of  their  imperfect  flexibility  are,  in  a  certain  degree, 
counteracted  by  bending  them  over  the  grooves  of  wheels. 
But  although  this  so  far  diminishes  these  effects  as  to  render 


CHAP.    XX.  STRENGTH    OP   MATERIALS.  229 

rapes  practically  useful,  yet  still,  in  calculating  the  powers  of 
machinery,  it  is  necessary  to  take  into  account  some  conse- 
quences of  the  rigidity  of  cordage,  which,  even  by  these 
means,  are  not  removed. 

To  explain  the  way  in  which  the  stiffness  of  a  rope  modi- 
fies the  operation  of  a  machine,  we  shall  suppose  it  bent  over 
a  wheel,  and  stretched  by  weights  A  B,  Jig.  182.,  at  its  ex- 
tremities. The  weights  A  and  B  being  equal,  and  acting  at 
C  and  D  in  opposite  ways,  balance  the  wheel.  If  the  weight 
A  receive  an  addition,  it  will  overcome  the  resistance  of  B, 
and  turn  the  wheel  in  the  direction  DEC.  Now,  for  the 
present,  let  us  suppose  that  the  rope  is  perfectly  inflexible ; 
the  wheel  and  weights  will  be  turned  into  the  position  repre- 
sented in  Jig.  183.  The  leverage  by  which  A  acts  will  be 
diminished,  and  will  become  O  F,  having  been  before  O  C ; 
and  the  leverage  by  which  B  acts  will  be  increased  to  O  G, 
having  been  before  O  D. 

But  the  rope,  not  being  inflexible,  will  yield  partially  to  the 
effects  of  the  weights  A  and  B,  and  the  parts  A  C  and  B  D 
will  be  bent  into  the  forms  represented  in  Jig.  184.  The 
form  of  the  curvature  which  the  rope  on  each  side  of  the 
wheel  receives  is  still  such  that  the  descending  weight  A 
works  with  a  diminished  leverage  F  O,  while  the  ascending 
weight  resists  it  with  an  increased  leverage  G  O.  Thus  so 
much  of  the  moving  power  is  lost,  by  the  stiffness  of  the 
rope,  as  is  necessary  to  compensate  this  disadvantageous 
change  in  the  power  of  the  machine; 


CHAPTER  XX. 

ON   THE    STRENGTH    OF    MATERIALS. 

(330.)  EXPERIMENTAL  inquiries  into  the  laws  which  regu 
late  the  strength  of  solid  bodies,  or  their  power  to  resist 
forces  variously  applied  to  tear  or  break  them,  are  obstructed 
by  practical  difficulties,  the  nature  and  extent  of  which  are 
so  discouraging,  that  few  have  ventured  to  encounter  them 
at  all,  and  still  fewer  the  steadiness  to  persevere  until  any 
result  showing  a  general  law  has  been  obtained.  These 
difficulties  arise,  partly  from  the  great  forces  which  must  be 
applied,  but  more  from  the  peculiar  nature  of  the  objects  of 
20 


230  THE    ELEMENTS    OF    MECHANICS.  CHAP.    XX 

those  experiments.  The  end  to  which  such  an  inquiry  must 
be  directed  is  the  developement  of  a  general  Irtw ;  that  is, 
such  a  rule  as  would  be  rigidly  observed  if  the  materials,  the 
strength  of  which  is  the  object  of  inquiry,  were  perfectly 
uniform  in  their  texture,  and  subject  to  no  casual  inequalities. 
In  proportion  as  these  inequalities  are  frequent,  experiments 
must  be  multiplied,  that  a  long  average  may  embrace  cases 
varying  in  both  extremes,  so  as  to  eliminate  each  other's 
effects  in  the  final  result. 

The  materials  of  which  structures  and  works  of  art  are 
composed  are  liable  to  so  many  and  so  considerable  inequali- 
ties of  texture,  that  any  rule  which  can  be  deduced,  even  by 
the  most  extensive  series  of  experiments,  must  be  regarded 
as  a  mean  result,  from  which  individual  examples  will  be 
found  to  vary  in  so  great  a  degree,  that  more  than  usual  cau- 
tion must  be  observed  in  its  practical  application.  The  de- 
tails- of  this  subject  belong  to  engineering  more  properly 
than  to  the  elements  of  mechanics.  Nevertheless,  a  general 
view  of  the  most  important  principles  which  have  been  es- 
tablished respecting  the  strength  of  materials  will  not  be 
misplaced  in  this  treatise. 

A  piece  of  solid  matter  may  be  submitted  to  the  action  of 
a  force  tending  to  separate  its  parts  in  several  ways  ;  the 
principal  of  which  are, — 

1.  To  a  direct  putt, — as  when  a  rope  or  wire  is  stretched 
by  a  weight ;  when  a  tie-beam  resists  the  separation  of  the 
sides  of  a  structure,  &,c. 

2.  To  a  direct  pressure  or  thrust,— as  when  a  weight  rests 
upon  a  pillar. 

3.  To  a  transverse  strain, — as  when  weights  on  the  ends 
of  a  lever  press  it  on  the  fulcrum. 

(331.)  If  a  solid  be  submitted  to  a  force  which  draws  it  in 
the  direction  of  its  length,  having  a  tendency  to  pull  its  ends 
in  opposite  directions,  its  strength  or  power  to  resist  such  a 
force  is  proportional  to  the  magnitude  of  its  transverse  section. 
Thus,  suppose  a  square  rod  of  metal  A  B,  Jig.  185.,  of  the 
breadth  and  thickness  of  one  inch,  be  pulled  by  a  force  in 
the  direction  A  B,  and  that  a  certain  force  is  found  sufficient 
to  tear  it ;  a  rod  of  the  same  metal  of  twice  the  breadth  and 
the  same  thickness  will  require  double  the  force  to  break  it ; 
one  of  treble  the  breadth  and  the  same  thickness  will  require 
treble  the  force  to  break  it ;  and  so  on. 

The  reason  of  this  is  evident.     A  rod  of  double  or  treble 


CHAP.  XX.  STRENGTH    OF    BEAMS.  231 

the  thickness,  in  this  case,  is  equivalent  to  two  or  three  equal 
and  similar  rods  which  equally  and  separately  resist  the  draw- 
ing force,  and  therefore  possess  a  degree  of  strength  pro- 
portionate to  their  number. 

It  will  easily  be  perceived,  that  whatever  be  the  section, 
the  same  reasoning  will  be  applicable,  and  the  power  of  re- 
sistance will,  in  general,  be  proportional  to  its  magnitude  or 
area. 

If  the  material  were  perfectly  uniform  throughout  its  di- 
mensions, the  resistance  to  a  direct  pull  would  not  be  affected 
by  the  length  of  the  rod.  In  practice,  however,  the  increase 
of  length  is  found  to  lessen  the  strength.  This  is  to  be  at- 
tributed to  the  increased  chance  of  inequality. 

(332.)  No  satisfactory  results  have  been  obtained  either 
by  theory  or  experiment  respecting  the  laws  by  which  solids 
resist  compression.  The  power  of  a  perpendicular  pillar  to 
support  a  weight  placed  upon  it,  evidently  depends  on  its 
thickness,  or  the  magnitude  of  its  base,  and  on  its  height. 
It  is  certain  that  when  the  height  is  the  same,  the  strength 
increases  with  every  increase  of  the  base ;  but  it  seems  doubt- 
ful whether  the  strength  be  exactly  proportional  to  the  base. 
That  is,  if  two  columns  of  the  same  material  have  equal 
heights,  and  the  base  of  one  be  double  the  base  of  the  other, 
the  strength  of  one  will  be  greater,  but  it  is  not  certain 
whether  it  will  exactly  double  that  of  the  other.  According 
to  the  theory  of  Euler,  which  is,  in  a  certain  degree,  verified 
by  the  experiments  of  Musschen brock,  the  strength  will  be 
increased  in  a  greater  proportion  than  the  base,  so  that  if  the 
base  be  doubled,  the  strength  will  be  more  than  doubled. 

When  the  base  is  the  same,  the  strength  is  diminished  by 
increasing  the  height,  and  this  decrease  of  strength  is  propor- 
tionally greater  than  the  increase  of  height.  According  to 
Euler's  theory,  the  decrease  of  strength  is  proportional  to  the 
square  of  the  height;  that  is,  when  the  height  is  increased 
in  a  two-fold  proportion,  the  strength  is  diminished  in  a  four- 
fold proportion. 

(333.)  The  strain  to  which  solids  forming  the  parts  of 
structures  of  every  kind  are  most  commonly  exposed,  is  the 
lateral  or  transverse  strain,  or  that  which  acts  at  right  angles 
to  their  lengths.  If  any  strain  act  obliquely  to  the  direction 
of  their  length,  it  may  be  resolved  into  two  forces  (76.),  one 
in  the  direction  of  the  length,  and  the  other  at  right  angles 
to  the  length.  That  part  which  acts  in  the  direction  of  the 


232  THE  ELEMENTS  OF  MECHANICS.      CHAP.  XX. 

length  will  produce  either  compression  or  a  direct  pull,  and 
its  effect  must  be  investigated  accordingly. 

Although  the  results  of  theory,  as  well  as  those  of  experi- 
mental investigations,  present  great  discordances  respecting 
the  transverse  strength  of  solids,  yet  there  are  some  particu- 
lars, in  which  they,  for  the  most  part,  agree  ;  to  these  it  is  our 
object  here  to  confine  our  observations,  declining  all  details 
relating  to  disputed  points. 

Let  A  B  C  D,  Jig.  186.,  be  a  beam,  supported  at  its  ends 
A  and  B.  Its  strength  to  support  a  weight  at  E,  pressing 
downwards  at  right  angles  to  its  length,  is  evidently  propor- 
tional to  its  breadth,  the  other  things  being  the  same.  For 
a  beam  of  double  or  treble  breadth,  and  of  the  same  thick- 
ness, is  equivalent  to  two  or  three  equal  and  similar  beams 
placed  side  by  side.  Since  each  of  these  would  possess  the 
same  strength,  the  whole  taken  together  would  possess  double 
or  treble  the  strength  of  any  one  of  them. 

When  the  breadth  and  length  are  the  same,  the  strength 
obviously  increases  with  the  depth,  but  not  in  the  same  pro- 
portion. The  increase  of  strength  is  found  to  be  much  greater 
in  proportion  than  the  increase  of  depth.  By  the  theory  of 
Galileo,  a  double  or  treble  thickness  ought  to  increase  the 
strength  in  a  four-fold  or  nine-fold  proportion,  and  experi- 
ments, in  most  cases,  do  not  materially  vary  from  this  rule. 

If,  while  the  breadth  and  depth  remain  the  same,  the  length 
of  the  beam,  or  rather  the  distance  between  the  points  of 
support,  vary,  the  strength  will  vary  accordingly,  decreasing 
in  the  same  proportion  as  the  length  increases. 

From  these  observations  it  appears,  that  the  transverse 
strength  of  a  beam  depends  more  on  its  thickness  than  its 
breadth.  Hence  we  find  that  a  broad  thin  board  is  much 
stronger  when  its  edge  is  presented  upwards.  On  this  prin- 
ciple the  joists  or  rafters  of  floors  and  roofs  are  constructed. 

If  two  beams  be  in  all  respects  similar,  their  strengths  will 
be  in  the  proportion  of  the  squares  of  their  lengths.  Let  the 
length,  breadth  and  depth  of  the  one  be  respectively  double 
the  length,  breadth  and  depth  of  the  other.  By  the  double 
breadth  the  beam  doubles  its  strength,  but  by  doubling  the 
length  half  this  strength  is  lost.  Thus  the  increase  of  length 
and  breadth  counteract  each  other's  effects,  and,  as  far  as  they 
are  concerned,  the  strength  of  the  beam  is  not  changed. 
But  by  doubling  the  thickness,  the  strength  is  increased  in  a 
four-fold  proportion,  that  is,  as  the  square  of  the  length.  In 


CHAP.  XX.      STRENGTH  OF  A  STRUCTURE.  233 

the  same  manner  it  may  be  shown,  that  when  all  the  dimen- 
sions are  trebled,  the  strength  is  increased  in  a  nine-fold 
proportion,  and  so  on. 

(334.)  In  all  structures  the  materials  have  to  support  their 
own  weight,  and  therefore  their  available  strength  is  to  be 
estimated  by  the  excess  of  their  absolute  strength  above  that 
degree  of  strength  which  is  just  sufficient  to  support  their 
own  weight.  This  consideration  leads  to  some  conclusions, 
of  which  numerous  and  striking  illustrations  are  presented 
in  the  works  of  nature  and  art. 

We  have  seen  that  the  absolute  strength  with  which  a  lat- 
eral strain  is  resisted  is  in  the  proportion  of  the  square  of  the 
linear  dimensions  of  similar  parts  of  a  structure,  and  there- 
fore the  amount  of  this  strength  increases  rapidly  with  every 
increase  of  the  dimensions  of  a  body.  But  at  the  same  time 
t!ie  wei^'it  of  the  body  increases  in  a  still  more  rapid  propor- 
tion. Thus,  if  the  several  dimensions  be  doubled,  the 
strength  will  be  increased  in  a  four-fold,  but  the  weight  in  an 
eight-fold  proportion.  If  the  dimensions  be  trebled,  the 
strength  will  be  multiplied  nine  times,  but  the  weight  twenty- 
seven  times.  Again,  if  the  dimensions  be  multiplied  four 
times,  the  strength  will  be  multiplied  sixteen  times,  and  the 
weight  sixty-four  times,  and  so  on. 

Hence  it  is  obvious,  that  although  the  strength  of  a  body 
of  small  dimensions  may  greatly  exceed  its  weight,  and, 
therefore,  it  may  be  able  to  support  a  load  many  times  its 
own  weight,  yet  by  a  great  increase  in  the  dimensions,  the 
weight  increasing  in  a  much  greater  degree,  the  available 
strength  must  be  much  diminished,  and  such  a  magnitude 
may  be  assigned,  that  the  weight  of  the  body  must  exceed  its 
strength,  and  it  not  only  would  be  unable  to  support  any  load, 
but  would  actually  fall  to  pieces  by  its  own  weight. 

The  strength  of  a  structure  of  any  kind  is  not,  therefore, 
to  be  determined  by  that  of  its  model,  which  will  always  be 
much  stronger  in  proportion  to  its  size.  All  works,  natural 
and  artificial,  have  limits  of  magnitude  which,  while  their 
materials  remain  the  same,  they  cannot  surpass. 

In  conformity  with  what  has  just  been  explained,  it  has 
been  observed,  that  small  animals  are  stronger  in  proportion 
than  large  ones;  that  the  young  plant  has  more  available 
strength  in  proportion  than  the  large  forest  tree ;  that  chil- 
dren are  less  liable  to  injury  from  accident  than  men,  &-c. 
But  although,  to  a  certain  extent,  these  observations  are  just, 
20* 


234  THE    ELEMENTS    OF    MECHANICS.  CHAP.    XXI 

yet  it  ought  not  to  be  forgotten,  that  the  mechanical  conclu- 
sions which  they  are  brought  to  illustrate  are  founded  on  the 
supposition,  that  the  smaller  and  greater  bodies  which  are 
compared  are  composed  of  precisely  similar  materials.  This 
is  not  the  case  in  any  of  the  examples  here  adduced. 


CHAPTER  XXI. 

ON    BALANCES    AND    PENDULUMS. 

335.)  THE  preceding  chapters  have  been  confined  almost 
wholly  to  the  consideration  of  the  laws  of  mechanics,  without 
entering  into  a  particular  description  of  the  machinery  and 
instruments  dependent  upon  those  laws.  Such  descriptions 
would  have  interfered  too  much  with  the  regular  progress  of 
the  subject,  and  it  therefore  appeared  preferable  to  devote  a 
chapter  exclusively  to  this  portion  of  the  work. 

Perhaps  there  are  no  ideas  which  man  receives  through 
the  medium  of  sense  which  may  not  be  referred  ultimately  to 
matter  and  motion.  In  proportion,  therefore,  as  he  becomes 
acquainted  with  the  properties  of  the  one  and  the  laws  of  the 
other,  his  knowledge  is  extended  ;  his  comforts  are  multiplied  ; 
lie  is  enabled  to  bend  the  powers  of  nature  to  his  will,  and 
to  construct  machinery  which  effects  with  ease  that  which 
the  united  labor  of  thousands  would  in  vain  be  exerted  to 
accomplish. 

Of  the  properties  of  matter,  one  of  the  most  important  is 
its  weight ;  and  the  element  which  mingles  inseparably  with 
the  laws  of  motion  is  time. 

In  the  present  chapter,  it  is  our  intention  to  describe  such 
instruments  as  are  usually  employed  for  determining  the 
weight  of  bodies.  To  attempt  a  description  of  the  various 
machines  which  are  used  for  the  measurement  of  time,  would 
lead  us  into  too  wide  a  field  for  the  present  occasion,  and 
we  shall,  therefore,  confine  ourselves  to  an  account  of  the 
methods  which  have  been  practised  to  perfect  that  instrument 
which  affords  the  most  correct  means  of  measuring  time — the 
pendulum. 

The  instrument  by  which  we  are  enabled  to  determine, 
with  greater  accuracy  than  by  any  other  means,  the  relative 


CHAP.    XXI.  THE    BALANCE.  235 

weight  of  a  body,  compared  with  the  weight  of  another  body 
assumed  as  a  standard,  is  the  balance. 

Of  the  Balance. 

The  balance  may  be  described  as  consisting  of  an  inflexi- 
ble rod  or  lever,  called  the  beam,  furnished  with  three  axes ; 
one,  the  fulcrum  or  centre  of  motion,  situated  in  the  middle, 
upon  which  the  beam  turns,  and  the  other  two  near  the  ex- 
tremities, and  at  equal  distances  from  the  middle.  These 
last  are  called  the  points  of  support,  and  serve  to  sustain  the 
pans  or  scales. 

The  points  of  support  and  the  fulcrum  are  in  the  same 
right  line,  and  the  centre  of  gravity  of  the  whole  should  be 
a  little  below  the  fulcrum  when  the  position  of  the  beam  is 
hodzont  il. 

The  arms  of  the  lever  being  equal,  it  follows  that  if  equal 
weights  be  put  into  the  scales,  no  effect  will  be  produced  on 
the  position  of  the  balance,  and  the  beam  will  remain  hori- 
zontal. 

If  a  small  addition  be  made  to  the  weight  in  one  of  the 
scales,  the  horizontality  of  the  beam  will  be  disturbed ;  and 
after  oscillating  for  some  time,  it  will,  on  attaining  a  state  of 
rest,  form  an  angle  with  the  horizon,  the  extent  of  which  is 
a  measure  of  the  delicacy  or  sensibility  of  the  balance. 

As  the  sensibility  of  a  balance  is  of  the  utmost  importance 
in  nice  scientific  inquiries,  we  shall  enter  somewhat  at  large 
into  a  consideration  of  the  circumstances  by  which  this  prop- 
erty is  influenced. 

In  Jig.  187.  let  A  B  represent  the  beam  drawn  from  the 
horizontal  position  by  a  very  small  weight  placed  in  the  scale 
suspended  from  the  point  of  support  B ;  then  the  force  tend- 
ing to  draw  the  beam  from  the  horizontal  position  may  be 
expressed  by  P  B  multiplied  by  such  very  small  weight  acting 
upon  the  point  B. 

Let  the  centre  of  gravity  of  the  whole  be  at  G  ;  then  the 
force  acting  against  the  former  will  be  G  P  multiplied  into 
the  weight  of  the  beam  and  scales,  and  when  these  forces 
are  equal,  the  beam  will  rest  in  arv  inclined  position.  Hence 
we  may  perceive  that  as  the  centre  of  gravity  is  nearer  to  or 
further  from  the  fulcrum  S,  (every  thing  else  remaining  the 
same,)  the  sensibility  of  the  balance  will  be  increased  or 
diminished. 


236  THE  ELEMENTS  OF  MECHANICS.     CHAP.  XXI. 

For,  suppose  the  centre  of  gravity  were  removed  to  g ; 
then,  to  produce  an  opposing  force  equal  to  that  acting  upon 
the  extremity  of  the  beam,  the  distance  g p  from  the  perpen- 
dicular line  must  be  increased  until  it  becomes  nearly  equal 
to  G  P  ;  but  for  this  purpose  the  end  of  the  beam  B  must 
descend,  which  will  increase  the  angle  II  S  B. 

As  all  weights  placed  in  the  scales  are  referred  to  the  line 
joining  the  points  of  support,  and  as  this  line  is  above  the 
centre  of  gravity  of  the  beam  when  not  loaded,  such  weights 
will  raise  the  centre  of  gravity ;  but  it  will  be  seen  that  the 
sensibility  of  the  balance,  as  far  as  it  depends  upon  this 
cause,  will  remain  unaltered. 

For,  calling  the  distance  S  G  unity,  the  distance  of  the 
centre  of  gravity  from  the  point  S  (to  which  the  weight 
which  has  been  added  is  referred)  will  be  expressed  by  the 
reciprocal  of  the  weight  of  the  beam  so  increased ;  that  is, 
if  the  weight  of  the  beam  be  doubled  by  weights  placed  in 
the  scales,  S  g  will  be  one  half  of  S  G ;  and  if  the  weight 
of  the  beam  be  in  like  manner  trebled,  S  g  will  be  one  third 
of  S  G,  and  so  on.  And  as  G  P  varies  as  S  G,  g  p  will  be 
inversely  proportionate  to  the  increased  weight  of  the  beam, 
and,  consequently,  the  product  obtained  by  multiplying  g  p 
by  the  weight  of  the  beam  and  its  load  will  be  a  constant 
quantity,  and  the  sensibility  of  the  balance,  as  before  stated, 
will  suffer  no  alteration. 

We  will  now  suppose  that  the  fulcrum  S,Jig.  188.,  is  situ- 
ated below  the  line  joining  the  points  of  support,  and  that 
the  centre  of  gravity  of  the  beam  when  not  loaded  is  at  G ; 
also  that  when  a  very  small  weight  is  placed  in  the  scale 
suspended  from  the  point  B,  the  beam  is  drawn  from  its  hor- 
izontal position,  the  deviation  being  a  measure  of  the  sensi- 
bility of  the  balance.  Then,  as  before  stated,  G  P  multiplied 
by  the  weight  of  the  beam  will  be  equal  to  P'  B  multiplied 
by  the  very  small  additional  weight  acting  on  the  point  B. 

Now,  if  we  place  equal  weights  in  both  scales,  such  addi- 
tional weights  will  be  referred  to  the  point  W,  and  the  result- 
ing distance  of  the  centre  of  gravity  from  the  point  W, 
calling  W  G  unity,  will  be  expressed  as  before  by  the  recip- 
rocal of  the  increased  weight  of  the  loaded  beam,  But  G  P 
will  decrease  in  a  greater  proportion  than  W  G :  thus,  sup- 
posing the  weight  of  the  beam  to  be  doubled,  W  g  would  be 
one  half  of  W  G ;  but  g  p,  as  will  be  evident  on  an  inspec- 
tion of  the  figure,  will  be  less  than  half  of  G  P ;  and  the 


CHAP.    XXI.  THE    BALANCE.  237 

same  small  weight  which  was  before  applied  to  the  point  B, 
if  now  added,  would  depress  the  point  B,  until  the  distance 
g  p  became  such  as  that,  when  multiplied  by  the  weight  of 
the  whole,  the  product  would  be  as  before  equal  to  P'  B  mul- 
tiplied by  the  before  mentioned  very  small  added  weight. 
The  sensibility  of  the  balance,  therefore,  in  this  case,  would 
be  increased. 

If  the  beam  be  sufficiently  loaded,  the  centre  of  gravity 
will  at  length  be  raised  to  the  fulcrum  S,  and  the  beam  will 
rest  indifferently  in  any  position.  If  more  weight  be  then 
added,  the  centre  of  gravity  will  be  raised  above  the  fulcrum, 
and  the  beam  will  turn  over. 

Lastly,  if  the  fulcrum  S,Jig.  189.,  is  above  the  line  joining 
the  two  points  of  support,  as  any  additional  weights  placed 
in  the  scales  will  be  referred  to  the  point  W,  in  the  line 
joining  A  and  B,  if  the  weight  of  the  beam  be  doubled  by 
such  added  weights,  and  the  centre  of  gravity  be  consequent- 
ly raised  to  g,  W  g  will  become  equal  to  half  of  W  G.  But 
g  p,  being  greater  than  one  half  of  G  P,  the  end  of  the 
beam  B  will  rise  until  g  p  becomes  such  as  to  be  equal,  when 
multiplied  by  the  whole  increased  weight  of  the  beam,  to 
P  B,  multiplied  by  the  small  weight  which  we  suppose  to 
have  been  placed  as  in  the  preceding  examples,  in  the 
scale. 

From  what  has  been  said,  it  will  be  seen  that  there  are 
three  positions  of  the  fulcrum  which  influence  the  sensibility 
of  the  balance ;  first,  when  the  fulcrum  and  the  points  of 
support  are  in  a  right  line,  when  the  sensibility  of  the  bal- 
ance will  remain  the  same,  though  the  weight  with  which 
the  beam  is  loaded  should  be  varied;  secondly,  when  the  ful- 
crum is  below  the  line  joining  the  two  points  of  support,  in 
which  case  the  sensibility  of  the  balance  will  be  increased  by 
additional  weights,  until  at  length  the  centre  of  gravity  is 
raised  above  the  fulcrum,  when  the  beam  will  turn  over ; 
and,  thirdly,  when  the  fulcrum  is  above  the  line  joining  the 
two  points  of  support,  in  which  case  the  sensibility  of  the 
balance  will  be  diminished  as  the  weight  with  which  the 
beam  is  loaded  is  increased. 

The  sensibility  of  a  balance,  as  here  defined,  is  the  angu- 
lar deviation  of  the  beam  occasioned  by  placing  an  additional 
constant  small  weight  in  one  of  the  scales  ;  but  it  is  frequent- 
ly expressed  by  the  proportion  which  such  small  additional 
weight  bears  to  the  weight  of  the  beam  and  its  load,  and 


238  THE  ELEMENTS  OF  MECHANICS.     CHAP.  XXI. 

sometimes  to  the  weight  the  value  of  which  is  to  be  deter- 
mined. 

This  proportion,  however,  will  evidently  vary  with  different 
weights,  except  in  the  case  where  the  centre  of  gravity  of  the 
beam  is  in  the  line  joining  the  points  supporting  the  scales,  the 
fulcrum  being  above  this  line  ;  and  it  is  therefore  necessary, 
in  every  other  case,  when  speaking  of  the  sensibility  of  the 
balance,  to  designate  the  weight  with  which  it  is  loaded ; 
thus,  if  a  balance  has  a  troy  pound  in  each  scale,  and  the 
horizontally  of  the  beam  varies  a  certain  small  quantity,  just 
perceptible  on  the  addition  of  one  hundredth  of  a  grain,  we 
say  that  the  balance  is  sensible  to  Ty-g^innr  part  of  its  load 
with  a  pound  in  each  scale,  or  that  it  will  determine  the 
weight  of  a  troy  pound  within  ^y^Voir  Part  °f  tne  whole. 

The  nearer  the  centre  of  gravity  of  a  balance  is  to  its 
fulcrum,  the  slower  will  be  the  oscillations  of  the  beam. 
The  number  of  oscillations,  therefore,  made  by  the  beam  in 
a  given  time  (a  minute  for  example),  affords  the  most  accu- 
rate method  of  judging  of  the  sensibility  of  the  balance, 
which  will  be  the  greater  as  the  oscillations  are  fewer. 

Balances  of  the  most  perfect  kind  (and  of  such  only  it  is 
our  present  object  to  treat)  are  usually  furnished  with  adjust- 
ments, by  means  of  which  the  length  of  the  arms,  or  the 
distances  of  the  fulcrum  from  the  points  of  support,  may  be 
equalized,  and  the  fulcrum  and  the  two  points  of  support  be 
placed  in  a  right  line ;  but  these  adjustments,  as  will  hereaf- 
ter be  seen,  are  not  absolutely  necessary. 

The  beam  is  variously  constructed,  according  to  the  pur- 
poses to  which  the  balance  is  to  be  applied.  Sometimes  it  is 
made  of  a  rod  of  solid  steel ;  sometimes  of  two  hollow  cones 
joined  at  their  bases ;  and,  in  some  balances,  the  beam  is  a 
frame  in  the  form  of  a  rhombus ;  the  principal  object  in  all, 
however,  is  to  combine  strength  and  inflexibility  with  light- 
ness. 

A  balance  of  the  best  kind,  made  by  Troughton,  is  so 
contrived  as  to  be  contained,  when  not  in  use,  in  a  drawer 
below  the  case  ;  and  when  in  use,  it  is  protected  from  any 
disturbance  from  currents  of  air,  by  being  enclosed  in  the 
case  above  the  drawer,  the  back  and  front  of  which  are  of 
plate  glass.  There  are  doors  in  the  sides,  through  which 
the  scale-pans  are  loaded,  and  there  is  a  door  at  the  top 
through  which  the  beam  may  be  taken  out. 

A  strong  brass  pillar,  in  the  centre  of  the  box,  supports  a 


CHAP.  xxi.  TROUGIITON'S  AND  ROBINSON'S  BALANCES.      239 

square  piece,  on  the  front  and  back  of  which  rise  two  arches, 
nearly  semicircular,  on  which  are  fixed  two  horizontal  planes 
of  agate,  intended  to  support  the  fulcrum.  Within  the  pillar 
is  a  cylindrical  tube,  which  slides  up  and  down  by  means  of 
a  handle  on  the  outside  of  the  case.  To  the  top  of  this  in- 
terior tube  is  fixed  an  arch,  the  terminations  of  which  pass 
beneath  and  outside  of  the  two  arches  before  described. 
These  terminations  are  formed  into  Y  5,  destined  to  receive 
the  ends  of  the  fulcrum,  which  are  made  cylindrical  for  this 
purpose,  when  the  interior  tube  is  elevated  in  order  to  relieve 
the  axis  when  the  balance  is  not  in  use.  On  depressing  the 
interior  tube,  the  Y  s  quit  the  axis,  and  leave  it  in  its  proper 
position  on  the  agate  planes.  The  beam  is  about  eighteen 
inches  long,  and  is  formed  of  two  hollow  cones  of  brass, 
joined  at  their  bases.  The  thickness  of  the  brass  does  not 
exceed  0-02  of  an  inch,  but  by  means  of  circular  rings  driven 
into  the  cones  at  intervals,  they  are  rendered  almost  inflexible. 
Across  the  middle  of  the  beam  passes  a  cylinder  of  steel,  the 
lower  side  of  which  is  formed  into  an  edge,  having  an  angle 
of  about  thirty  degrees,  which,  being  hardened  and  well  pol- 
ished, constitutes  the  fulcrum,  and  rests  upon  the  agate 
planes  for  the  length  of  about  0-05  of  an  inch. 

Each  point  of  suspension  is  formed  of  an  axis  having  two 
sharp  concave  edges,  upon  which  rest  at  right  angles  two 
other  sharp  concave  edges  formed  in  the  spur-shaped  piece  to 
which  the  strings  carrying  the  scale-pan  are  attached.  The 
two  points  are  adjustable,  the  one  horizontally,  for  the  pur- 
pose of  equalizing  the  arms  of  the  beam,  and  the  other  ver- 
tically, for  bringing  the  points  of  suspension  and  the  fulcrum 
into  a  right  line. 

Such  is  the  form  of  Troughton's  balance.  We  shall  now 
give  the  description  of  a  balance  as  constructed  by  Mr.  Rob- 
inson of  Devonshire  Street,  Portland  Place  : — 

The  beam  of  this  balance  is  only  ten  inches  long.  It  is  a 
frame  of  bell-metal  in  the  form  of  a  rhombus.  The  fulcrum 
is  an  equilateral  triangular  prism  of  steel  one  inch  in  length ; 
but  the  edge  on  which  the  beam  vibrates  is  formed  to  an 
angle  of  120°,  in  order  to  prevent  any  injury  from  the  weight 
with  which  it  may  be  loaded.  The  chief  peculiarity  in  this 
balance  consists  in  the  knife-edge  which  forms  the  fulcrum 
bearing  upon  an  agate  plane  throughout  its  whole  length, 
whereas  we  have  seen  in  the  balance  before  described  that 
the  whole  weight  is  supported  by  portions  only  of  the  knife- 


240  THE    ELEMENTS    OP    MECHANICS.  CHAP.  XXI 

edge,  amounting  together  to  one  tenth  of  an  inch.  The  sup- 
ports for  the  scales  are  knife-edges,  each  six  tenths  of  an  inch 
long.  These  are  each  furnished  with  two  pressing  screws, 
by  means  of  which  they  may  be  made  parallel  to  the  central 
knife-edge. 

Each  end  of  the  beam  is  sprung  obliquely  upwards  and 
towards  the  middle,  so  as  to  form  a  spring  through  which  a 
pushing  screw  passes,  which  serves  to  vary  the  distance  of 
the  point  of  support  from  the  fulcrum,  and,  at  the  same  time, 
by  its  oblique  action,  to  raise  or  depress  it,  so  as  ,o  furnish  a 
means  of  bringing  the  points  of  support  and  the  fulcrum  into 
a  right  line. 

A  piece  of  wire,  four  inches  long,  on  which  a  screw  is  cut, 
proceeds  from  the  middle  of  the  beam  downvyards.  This  is 
pointed  to  serve  as  an  index,  and  a  small  brass  ball  moves  on 
the  screw,  by  changing  the  situation  of  which  the  place  of 
the  centre  of  gravity  may  be  varied  at  pleasure. 

The  fulcrum,  as  before  remarked,  rests  upon  an  agate  plane 
throughout  its  whole  length,  and  the  scale-pans  are  attached 
to  planes  of  agate  which  rest  upon  the  knife-edges  forming 
the  points  of  support.  This  method  of  supporting  the  scale- 
pans,  we  have  reason  to  believe,  is  due  to  Mr.  Cavendish. 
Upon  the  lower  half  of  the  pillar  to  which  the  agate  plane  is 
fixed,  a  tube  slides  up  and  down  by  means  of  a  lever  which 
passes  to  the  outside  of  the  case.  From  the  top  of  this  tube 
arms  proceed  obliquely  towards  the  ends  of  the  balance,  serv- 
ing to  support  a  horizontal  piece,  carrying  at  each  extremity 
two  sets  of  Y  s,  one  a  little  above  the  other.  The  upper  Y  5 
are  destined  to  receive  the  agate  planes  to  which  the  scale- 
pans  are  attached,  and  thus  to  relieve  the  knife-edges  from 
their  pressure ;  the  lower  to  receive  the  knife-edges  which 
form  the  points  of  support,  consequently  these  latter  Y  s, 
when  in  action,  sustain  the  whole  beam. 

When  the  lever  is  freed  from  a  notch  in  which  it  is  lodged, 
a  spring  is  allowed  to  act  upon  the  tube  we  have  mentioned, 
and  to  elevate  it.  The  upper  Y  s  first  meet  the  agate  planes 
carrying  the  scale-pans,  and  free  them  from  the  knife-edges. 
The  lower  Y  s  then  come  into  action,  and  raise  the  whole 
beam,  elevating  the  central  knife-edge  above  the  agate  plane. 
This  is  the  usual  state  of  the  balance  when  not  in  use  :  when 
it  is  to  be  brought  into  action,  the  reverse  of  what  we  have 
described  takes  place.  On  pressing  down  the  lever,  the  cen- 
tral knife-edge  first  meets  the  agate  plane,  and  afterwards  the 


CHAP.  xxi.  KATER'S  BALANCE.  241 

two  agate  planes  carrying  the  scale-pans  are  deposited  upon 
their  supporting  knife-edges. 

A  balance  of  this  construction  was  employed  by  the  writer 
of  this  article  in  adjusting  the  national  standard  pound.  With 
a  pound  troy  in  each  scale,  the  addition  of  one  hundreth  of  a 
grain  caused  the  index  to  vary  one  division,  equal  to  one  tenth 
of  an  inch,  and  Mr.  Robinson  adjusts  these  balances  so  that 
with  one  thousand  grains  in  each  scale,  the  index  varies  per- 
ceptibly on  the  addition  of  one  thousandth  of  a  grain,  or  of 
one  millionth  part  of  the  weight  to  be  determined. 

It  may  not  be  uninteresting  to  subjoin,  from  the  Philo- 
sophical Transactions  for  1826,  the  description  of  a  balance 
perhaps  the  most  sensible  that  has  yet  been  made,  construct- 
ed for  verifying  the  national  standard  bushel.  The  author 
says,— 

"  The  weight  of  the  bushel  measure,  together  with  the  80 
Ibs.  of  water  it  should  contain,  was  about  250  Ibs. ;  and  as  I 
could  find  no  balance  capable  of  determining  so  large  a  weight 
with  sufficient  accuracy,  I  was  under  the  necessity  of  con- 
structing one  for  this  express  purpose. 

"  I  first  tried  cast  iron  ;  but  though  tke  beam  was  made  as 
light  as  was  consistent  with  the  requisite  degree  of  strength, 
the  inertia  of  such  aMiass  appeared  to  be  so  considerable, 
that  much  time  must  have  been  lost  before  the  balance  would 
have  answered  to  the  small  differences  I  wished  to  ascertain. 
Lightness  was  a  property  essentially  necessary,  and  bulk  was 
very  desirable,  in  order  to  preclude  such  errors  as  might  arise 
from  the  beam  being  partially  affected  by  sudden  alterations 
of  temperature.  I  therefore  determined  to  employ  wood,  a 
material  in  which  the  requisites  I  sought  were  combined. 
The  beam  was  made  of  a  plank  of  mahogany,  about  70 
inches  long,  22  inches  wide,  and  2£  thick,  tapering  from  the 
middle  to  the  extremities.  An  opening  was  cut  in  the  centre, 
and  strong  blocks  screwed  to  each  side  of  the  plank,  to  form 
a  bearing  for  the  back  of  a  knife-edge  which  passed  through 
the  centre.  Blocks  were  also  screwed  to  each  side  at  the 
extremities  of  the  beam,  on  which  rested  the  backs  of  the 
knife-edges  for  supporting  the  pans.  The  opening  in  the 
centre  was  made  sufficiently  large  to  admit  the  support 
hereafter  to  be  described,  upon  which  the  knife-edge  rested. 

"  In  all  beams  which  I  have  seen,  with  the  exception  of 
those  made  by  Mr.  Robinson,  the  whole  weight  is  sustained 
by  short  portions  at  the  extremities  of  the  knife-edge ;  and 
21 


242  THE    ELEMENTS    OF    MECHANICS.  CHAP.    XXI. 

the  weight  being  thus  thrown  ftpon  a  few  points,  the  knife- 
edge  becomes  more  liable  to  change  its  figure  and  to  suffer 
injury. 

"  To  remedy  this  defect,  the  central  knife-edge  of  the  beam 
I  am  describing  was  made  6  inches,  and  the  two  others  5 
inches  long.  They  were  triangular  prisms  with  equal  sides  of 
three  fourths  of  an  inch,  very  carefully  finished,  and  the  edges 
ultimately  formed  to  an  angle  of  120°. 

"  Each  knife-edge  was  screwed  to  a  thick  plate  of  brass, 
the  surfaces  in  contact  having  been  previously  ground  togeth- 
er ;  and  these  plates  were  screwed  to  the  beam,  the  knife- 
edges  being  placed  in  the  same  plane,  and  as  nearly  equidis- 
tant and  parallel  to  each  other  as  could  be  done  by  construction. 

"  The  support  upon  which  the  central  knife-edge  rested 
throughout  its  whole  length  was  formed  of  a  plate  of  polished 
hard  steel,  screwed  to  a  block  of  cast  iron.  This  block  was 
passed  through  the  opening  before  mentioned  in  the  centre 
of  the  beam,  and  properly  attached  to  a  frame  of  cast  iron. 

"  The  stirrups  to  which  the  scales  were  hooked,  rested 
upon  plates  of  polished  steel  to  which  they  were  attached, 
and  the  under  surfaces  of  which  were  formed  by  careful  grind- 
ing into  cylindrical  segments.  These  were  in  contact  with 
the  knife-edges  their  whole  length,  and  were  known  to  be  in 
their  proper  position  by  the  correspondence  of  their  extremi- 
ties with  those  of  the  knife-edges.  A  well  imagined  con- 
trivance was  applied  by  Mr.  Bate  for  raising  the  beam  when 
loaded,  in  order  to  prevent  unnecessary  wear  of  the  knife- 
edge,  and  for  the  purpose  of  adjusting  the  place  of  the  centre 
of  gravity,  when  the  beam  was  loaded  with  the  weight  re- 
quired to  be  determined,  a  screw  carrying  a  movable  ball  pro- 
jected vertically  from  the  middle  of  the  beam. 

"  The  performance  of  this  balance  fully  equalled  my  ex- 
pectations. With  two  hundred  and  fifty  pounds  in  each  scale, 
the  addition  of  a  single  grain  occasioned  an  immediate  varia- 
tion in  the  index  of  one  twentieth  of  an  inch,  the  radius  being 
fifty  inches." 

From  the  preceding  account,  it  appears  that  this  balance  is 
sensible  to  yTF&innr  Part  °f  tne  weight  which  was  to  be  de- 
termined. 

We  shall  now  describe  the  method  to  be  pursued  in  adjust- 
ing a  balance. 

1.  To  bring  the  points  of  suspension  and  the  fulcrum  into 
a  right  line. 


CHAP.    XXI.  USE    OF    THE    BALANCE.  243 

Make  the  vibrations  of  the  balance  very  slow,  by  moving 
the  weight  which  influences  the  centre  of  gravity,  and  bring 
the  beam  into  a  horizontal  position,  by  means  of  small  bits 
of  paper  thrown  into  the  scales.  Then  load  the  scales  with 
nearly  the  greatest  weight  the  beam  is  fitted  to  carry.  If  the 
vibrations  are  performed  in  the  same  time  as  before,  no  fur- 
ther adjustment  is  necessary  ;  but  if  the  beam  vibrates  quick- 
er, or  if  it  oversets,  cause  it  to  vibrate  in  the  same  time  as  at 
first,  by  moving  the  adjusting  weight,  and  note  the  distance 
through  which  the  weight  has  passed.  Move  the  weight  then 
in  the  contrary  direction  through  double  this  distance,  and 
then  produce  the  former  slow  motion  by  means  of  the  screw 
acting  vertically  on  the  point  of  support.  Repeat  this  opera- 
tion until  the  adjustment  is  perfect. 

2.  To  make  the  arms  of  the  beam  of  an  equal  length. 

Put  weights  in  the  scales  as  before;  bring  the  beam  as 
nearly  as  possible  to  a  horizontal  position,  and  note  the  divis- 
ion at  which  the  index  stands ;  unhook  the  scales,  and  trans- 
fer them  with  their  weights  to  the  other  ends  of  the  beam, 
when,  if  the  index  points  to  the  same  division,  the  arms  are 
of  an  equal  length  ;  but  if  not,  bring  the  index  to  the  division 
which  had  been  noted,  by  placing  small  weights  in  one  or  the 
other  scale.  Take  away  half  these  weights,  and  bring  the 
index  again  to  the  observed  division  by  the  adjusting  screw, 
which  acts  horizontally  on  the  point  of  support.  If  the  scale- 
pans  are  known  to  be  of  the  same  weight,  it  will  not  be  ne- 
cessary to  change  the  scales,'but  merely  to  transfer  the  weights 
from  one  scale-pan  to  the  other. 

Of  the  Use  of  the  Balance. 

Though  we  have  described  the  method  of  adjusting  the 
balance,  these  adjustments,  as  we  have  before  remarked,  may 
be  dispensed  with.  Indeed,  in  all  delicate  scientific  opera- 
tions, it  is  advisable  never  to  rely  upon  adjustments,  which, 
after  every  care  has  been  employed  in  effecting  them,  can 
only  be  considered  as  approximations  to  the  truth.  We  shall, 
therefore,  now  describe  the  best  method  of  ascertaining  the 
weight  of  a  body,  and  which  does  not  depend  on  the  accura- 
cy of  these  adjustments. 

Having  levelled  the  case  which  contains  the  balance,  and 
thrown  the  beam  out  of  action,  place  a  weight  in  each  scale- 
pan  nearly  equal  to  the  weight  which  is  to  be  determined. 


244  THE    ELEMENTS    OF    MECHANICS.  CHAP.    XXI. 

Lower  the  beam  very  gently  till  it  is  in  action,  and,  by  means 
of  the  adjustment  for  raising  or  lowering  the  centre  of  gravi- 
ty, cause  the  beam  to  vibrate  very  slowly.  Remove  these 
weights,  and  place  the  substance,  the  weight  of  which  is  to 
be  determined,  in  one  of  the  scale-pans;  carefully  counter- 
pose  it  by  means  of  any  convenient  substances  put  into  the 
other  scale-pan,  and  observe  the  division  at  which  the  index 
stands ;  remove  the  body,  the  weight  of  which  is  to  be  ascer- 
tained, and  substitute  standard  weights  for  it  so  as  to  bring 
the  index  to  the  same  division  as  before.  These  weights  will 
be  equal  to  the  weight  of  the  body. 

If  it  be  required  to  compare  two  weights  together  which 
are  intended  to  be  equal,  and  to  ascertain  their  difference,  if 
any,  the  method  of  proceeding  will  be  nearly  the  same.  The 
standard  weight  is  to  be  carefully  counterpoised,  and  the  divis- 
ion at  which  the  index  stands,  noted.  And  now  it  will  be 
convenient  to  add  in  either  of  the  scales  some  small  weight, 
such  as  one  or  two  hundredths  of  a  grain,  and  mark  the  num- 
ber of  divisions  passed  over  in  consequence  by  the  index,  by 
which  the  value  of  one  division  of  the  scale  will  be  known. 
This  should  be  repeated  a  few  times,  and  the  mean  taken  for 
greater  certainty. 

Having  noted  the  division  at  which  the  index  rests,  the 
standard  weight  is  to  be  removed,  and  the  weight  which  is  to 
be  compared  with  it  substituted  for  it.  The  index  is  then 
again  to  be  noted,  and  the  difference  between  this  and  the 
former  indication  will  give  the  difference  between  the  weights 
in  parts  of  a  grain. 

If  the  balance  is  adjusted  so  as  to  be  very  sensible,  it  will 
be  long  before  it  comes  to  a  state  of  rest.  It  may,  therefore, 
sometimes  be  advisable  to  take  the  mean  of  the  extent  of  the 
vibrations  of  the  index  as  the  point  where  it  would  rest,  and 
this  may  be  repeated  several  times  for  greater  accuracy.  It 
must,  however,  be  remembered,  that  it  is  not  safe  to  do  this 
when  the  extent  of  the  vibrations  is  beyond  one  or  two  divis- 
ions of  the  scale ;  but  with  this  limitation,  it  is,  perhaps,  as 
good  a  method  as  can  be  pursued. 

Many  precautions  are  necessary  to  ensure  a  satisfactory 
result.  The  weights  should  never  be  touched  by  the  hand ; 
for  not  only  would  this  oxydate  the  weight,  but  by  raising  its 
temperature  it  would  appear  lighter,  when  placed  in  the  scale 
pan,  than  it  should  do,  in  consequence  of  the  ascent  of  the 
liented  air.  For  the  larger  weights,  a  wooden  fork  or  tongs, 


CHAP.  XXI.  WEIGHTS.  245 

according  to  the  form  of  the  weight,  should  be  employed ; 
•and  for  the  smaller,  a  pair  of  forceps  made  of  copper  will  be 
found  the  most  convenient ;  this  metal  possessing  sufficient 
elasticity  to  open  the  forceps  on  their  being  released  from 
pressure,  and  yet  not  opposing  a  resistance  sufficient  to  in- 
terfere with  that  delicacy  of  touch  which  is  desirable  in  such 
operations. 

Of  Weights. 

It  must  be  obvious,  that  the  excellence  of  the  balance  would 
be  of  little  use,  unless  the  weights  employed  were  equally  to 
be  depended  upon.  The  weights  may  either  be  accurately 
adjusted,  or  the  difference  between  each  weight  and  the 
standard  may  be  determined,  and,  consequently,  its  true 
value  ascertained.  It  has  been  already  shown  how  the  latter 
may  be  effected,  in  the  instructions  which  have  been  given 
for  comparing  two  weights  together ;  and  we  shall  now  show 
the  readiest  mode  of  adjusting  weights  to  an  exact  equality  with 
a  given  standard. 

The  material  of  the  weight  may  be  either  brass  or  platina, 
and  its  form  may  be  cylindrical ;  the  diameter  being  nearly 
twice  the  height.  A  small  spherical  knob  is  screwed  into 
the  centre,  a  space  being  left  under  the  screw  to  receive  the 
portions  of  fine  wire  used  in  the  adjustment.  It  will  be  con- 
venient to  form  a  cavity  in  the  bottom  of  each  weight,  to  re- 
ceive the  knob  of  the  weight  upon  which  it  may  be  placed. 

Each  weight  is  now  to  be  compared  with  the  standard,  and 
should  it  be  too  heavy,  it  is  to  be  reduced  till  it  becomes  in  a 
very  small  degree  too  light,  when  the  amount  of  the  deficien- 
cy is  to  be  carefully  determined. 

Some  very  fine  silver  wire  is  now  to  be  taken,  and  the 
weight  of  three  or  four  feet  of  it  ascertained.  From  this  it 
will  be  known  what  length  of  the  wire  is  equal  to  the  error 
of  the  weight  to  be  adjusted;  and  this  length  being  cut  off 
is  to  be  enclosed  under  the  screw.  To  guard  against  any 
possible  error,  it  will  be  advisable,  before  the  screw  is  firm- 
ly fixed  in  its  place,  again  to  compare  the  weight  with  the 
standard. 

The  most  approved  method  of  making  weights  expressing 
the  decimal  parts  of  a  grain,  is  to  determine,  as  before,  with 
great  care,  the  weight  of  a  certain  length  of  fine  wire,  and 
then  to  cut  off  such  portions  as  are  equal  to  the  weights  re- 
quired. 

21  * 


246  THE    ELEMENTS    OF    MECHANICS.  CHAP.  XXI. 

Before  we  conclude  this  article,  we  shall  give  a  description, 
from  the  Annals  of  Philosophy  for  1825,  of  "  a  very  sensible 
balance,"  used  by  the  late  Dr.  Black  : — 

"  A  thin  piece  of  fir  wood,  not  thicker  than  a  shilling,  and 
a  foot  long,  three  tenths  of  an  inch  broad  in  the  middle,  and 
one  tenth  and  a  half  at  each  end,  is  divided  by  transverse 
lines  into  twenty  parts ;  that  is,  ten  parts  on  each  side  of  the 
middle.  These  are  the  principal  divisions,  and  each  of  them 
is  subdivided  into  halves  and  quarters.  Across  the  middle  is 
fixed  one  of  the  smallest  needles  1  could  procure,  to  serve  as 
an  axis,  and  it  is  fixed  in  its  place  by  means  of  a  little  sealing 
wax.  The  numeration  of  the  divisions  is  from  the  middle  to 
each  end  of  the  beam.  The  fulcrum  is  a  bit  of  plate  bra^s, 
the  middle  of  which  lies  flat  on  my  table  when  I  use  the  bal- 
ance, and  the  two  ends  are  bent  up  to  a  right  angle  so  as  to 
stand  upright.  These  two  ends  are  ground  at  the  same  time 
on  a  flat  hone,  that  the  extreme  surfaces  of  them  may  be  in  the 
same  plane ;  and  their  distance  is  such  that  the  needle,  when 
laid  across  them,  rests  on  them  at  a  small  distance  from  the 
sides  of  the  beam.  They  rise  above  the  surface  of  the  table 
only  one  tenth  and  a  half,  or  two  tenths  of  an  inch,  so  that 
the  beam  is  very  limited  in  its  play.  See  Jig.  190. 

"  The  weights  I  use  are  one  globule  of  gold,  which  weighs 
one  grain,  and  two  or  three  others  which  weigh  one  tenth  of 
a  grain  each ;  and  also  a  number  of  small  rings  of  fine  brass 
wire,  made  in  the  manner  first  mentioned  by  Mr.  Lewis,  by 
appending  a  weight  to  the  wire,  and  coiling  it  with  the  ten- 
sion of  that  weight  round  a  thicker  brass  wire  in  a  close  spiral, 
after  which,  the  extremity  of  the  spiral  being  tied  hard  with 
waxed  thread,  I  put  the  covered  wire  into  a  vice,  and  apply- 
ing a  sharp  knife,  which  is  struck  with  a  hammer,  I  cut 
through  a  great  number  of  the  coils  at  one  stroke,  and  find 
them  as  exactly  equal  to  one  another  as  can  be  desired.  Those 
I  use  happen  to  be  the  ^  part  of  a  grain  each,  or  300  of 
them  weigh  ten  grains ;  but  I  have  others  much  lighter. 

"  You  will  perceive  that,  by  means  of  these  weights  placed 
on  different  parts  of  the  beam,  I  can  learn  the  weight  of  any 
little  mass  from  one  grain,  or  a  little  more,  to  the  T5-Vir  °f  a 
grain.  For  if  the  thing  to  be  weighed  weighs  one  grain,  it 
will,  when  placed  on  one  extremity  of  the  beam,  counterpoise 
the  large  gold  weight  at  the  other  extremity.  If  it  weighs 
half  a  grain,  it  will  counterpoise  the  heavy  gold  weight  placed 
at  5.  If  it  weigh  T6a  of  a  grain,  you  must  place  the  heavy  gold 


CHAP.  XXT.  DR.  BLACK'S  BALANCE.  247 

weight  at  5,  and  one  of  the  lighter  ones  at  the  extremity  to 
counterpoise  it ;  and  if  it  weighs  only  one  or  two,  or  three  or 
four  hundredths  of  a  grain,  it  will  be  counterpoised  by  one  of 
the  small  gold  weights  placed  at  the  first  or  second,  or  third 
or  fourth  division.  If,  on  the  contrary,  it  weighs  one  grain 
and  a  fraction,  it  will  be  counterpoised  by  the  heavy  gold 
weight  at  the  extremity,  and  one  or  more  of  the  lighter  ones 
placed  in  some  other  part  of  the  beam. 

"  This  beam  has  served  me  hitherto  for  every  purpose ;  but 
had  I  occasion  for  a  more  delicate  one,  I  could  make  it  easily 
by  taking  a  much  thinner  and  lighter  slip  of  wood,  and  grind- 
ing the  needle  to  give  it  an  edge.  It  would  also  be  easy  to 
make  it  carry  small  scales  of  paper  for  particular  purposes." 

The  writer  of  this  article  has  used  a  balance  of  this  kind, 
and  finds  that  it  is  sensible  to  ^JTT  °f  a  grain  when  loaded 
with  ten  grains.  It  is  necessary,  howerer,  where  accuracy 
is  required,  to  employ  a  scale-pan.  This  may  be  made  of 
thin  card  paper,  shaped  as  in  Jig.  191. 

A  thread  is  to  be  passed  through  the  two  ends,  by  tighten- 
ing which  they  may  be  brought  near  each  other. 

The  most  convenient  weights  for  this  beam  appear  to  be 
two  of  one  grain  each,  and  one  of  one  tenth  of  a  grain. 
They  .should  be  made  of  straight  wire ;  and  if  the  beam  be 
notched  at  the  divisions,  they  may  be  lodged  in  these  notches 
very  conveniently.  Ten  divisions  on  each  side  of  the  middle 
will  be  sufficient.  The  weight  of  the  scale-pan  must  first  be 
carefully  ascertained,  in  order  that  it  may  be  deducted  from 
the  weight,  afterwards  determined,  of  the  scale-pan  and  the 
substance  it  may  contain. 

If  the  scale-pan  be  placed  at  the  tenth  division  of  the 
beam,  it  is  evident  that  by  means  of  the  two  grain  weights, 
a  greater  weight  cannot  be  determined  than  one  grain  and 
nine  tenths  ;  but  if  the  scale-pan  be  placed  at  any  other 
division  of  the  beam,  the  resulting  apparent  weight  must  be 
increased  by  multiplying  it  by  ten,  and  dividing  by  the  num- 
ber of  the  division  at  which  the  scale-pan  is  placed  ;  and 
in  this  manner  it  is  evident  that  if  the  scale-pan  be  placed  at 
the  division  numbered  1,  a  weight  amounting  to  nineteen 
grains  may  be  determined. 

We  have  been  tempted  to  describe  this  little  apparatus 
because  it  is  extremely  simple  in  its  construction,  may  be 
easily  made,  and  may  be  very  usefully  employed  on  many 
occasions  where  extreme  accuracy  is  not  necessary. 


248  THE  ELEMENTS  OF  MECHANICS.     CHAP.  XXI. 

Description  of  the  Steelyard. 

Thn  steelyard  is  a  lever,  having  unequal  arms  ;  and  in  its 
most  simple  form  it  is  so  arranged,  that  one  weight  alone 
serves  to  determine  a  great  variety  of  others,  by  sliding  it 
along  the  longer  arm  of  the  lever,  and  thus  varying  its  dis- 
tance from  the  fulcrum. 

It  has  been  demonstrated,  Chapter  XIII.,  that  in  the  lever 
the  proportion  of  the  power  to  the  weight  will  be  always  the 
same  as  that  of  their  distances  from  the  fulcrum,  taken  in  a 
reverse  order  ;  consequently,  when  a  constant  weight  is  used, 
and  an  equilibrium  established  by  sliding  this  weight  on  the 
longer  arm  of  the  lever,  the  relative  weight  of  the  substance 
weighed,  to  the  constant  weight,  will  be  in  the  same  propor- 
tion as  the  distance  of  the  constant  weight  from  the  fulcrum 
is  to  the  length  of  the  shorter  arm. 

Thus,  suppose  the  length  of  the  shorter  arm,  or  the  distance 
of  the  fulcrum  from  the  point  from  which  the  weight  to  be 
determined  is  suspended,  to  be  one  inch  ;  let  the  longer  arm 
of  the  lever  be  divided  into  parts  of  one  inch  each,  begin- 
ning at  the  fulcrum.  Now  let  the  constant  weight  be  equal 
to  one  pound,  and  let  the  steelyard  be  so  constructed  that 
the  shorter  arm  shall  be  sufficiently  heavy  to  counterpoise 
the  longer  when  the  bar  is  unloaded.  Then  suppose  a  sub- 
stance, the  weight  of  which  is  five  pounds,  to  be  suspended 
from  the  shorter  arm.  It  will  be  found  that  when  the  con- 
stant weight  is  placed  at  the  distance  of  five  inches  from  the 
fulcrum,  the  weights  will  be  in  equilibrium,  and  the  bar 
consequently  horizontal.  In  this  steelyard,  therefore,  the 
distance  of  each  inch  from  the  fulcrum  indicates  a  weight 
of  one  pound.  An  instrument  of  this  form  was  used  by  the 
Romans,  and  it  is  usually  described  as  the  Roman  statera  or 
steelyard.  A  representation  of  it  is  given  at  Jig.  192. 

The  steelyard  is  in  very  general  use  for  the  coarser  pur- 
poses of  commerce,  but  constructed  differently  from  that 
which  we  have  described.  The  beam  with  the  scales  or 
hooks  is  seldom  in  equilibrium  upon  the  point  F,  when  the 
weight  P  is  removed  ;  but  the  longer  arm  usually  preponder- 
ates, and  the  commencement  of  the  graduations,  therefore, 
is  not  at  F,  but  at  some  point  between  B  and  F.  The  com- 
mon steelyard,  which  we  have  represented  at  Jig.  193.,  is 
usually  furnished  with  two  points,  from  either  of  which  the 
substance,  the  weight  of  which  is  to  be  determined,  may  be 


CHAP.  xxi.  c.  PAUL'S  STEELYARD.  249 

suspended.  The  value  of  the  divisions  is  in  this  case  in- 
creased in  proportion  as  the  length  of  the  shorter  arm  is 
decreased.  Thus,  in  the  steelyard  which  we  have  described, 
if  there  be  a  second  point  of  suspension  at  the  distance 
of  half  an  inch  from  the  fulcrum,  each  division  of  the  longer 
arm  will  indicate  two  pounds  instead  of  one,  and  these  di- 
visions are  usually  marked  upon  the  opposite  edge  of  the 
steelyard,  which  is  made  to  turn  over. 

This  instrument  is  very  convenient,  because  it  requires 
but  one  weight ;  and  the  pressure  on  the  fulcrum  is  less  than 
in  the  balance,  when  the  substance  to  be  weighed  is  heavier 
than  the  constant  weight.  But,  on  the  contrary,  when  the 
constant  weight  exceeds  the  substance  to  be  weighed,  the 
pressure  on  the  fulcrum  is  greater  in  the  steelyard  than  in 
the  balance,  and  the  balance  is,  therefore,  preferable  in  de- 
termining small  weights.  There  is  also  an  advantage  in  the 
balance,  because  the  subdivision  of  weights  can  be  effected 
with  a  greater  degree  of  precision  than  the  subdivision  of 
the  arm  of  the  steelyard. 

C.  Pau?s  Steelyard. 

A  steelyard  has  been  constructed  by  Mr.  C.  Paul,  inspector 
of  weights  and  measures  at  Geneva,  which  is  much  to  be  pre- 
ferred to  that  in  common  use.  Mr.  C.  Paul  states,  that  steel- 
yards have  two  advantages  over  balances  :  1.  That  their  axis 
of  suspension  is  not  loaded  with  any  other  weight  than  that 
of  the  merchandise,  the  constant  weight  of  the  apparatus 
itself  excepted ;  while  the  axis  of  the  balance,  besides  the 
weight  of  the  instrument,  sustains  a  weight  double  to  that  of 
the  merchandise.  2.  The  use  of  the  balance  requires  a  con- 
siderable assortment  of  weights,  which  causes  a  proportional 
increase  in  the  price  of  the  apparatus,  independently  of  the 
chances  of  error  which  it  multiplies,  and  of  the  time  employ- 
ed in  producing  an  equilibrium. 

1.  In  C.  Paul's  steelyard,  the  centres  of  the  movement  of 
suspension,  or  the  two  constant  centres,  are  placed  on  the 
exact  line  of  the  divisions  of  the  beam ;  an  elevation  almost 
imperceptible  in  the  axis  of  the  beam,  destined  to  compensate 
for  the  very  slight  flexion  of  the  bar,  alone  excepted. 

2.  The  apparatus,  by  the  construction  of  the  beam,  is  bal- 
anced below  its  centre  of  motion,  so  that  when  no  weight  is 
suspended,  the  beam  naturally  remains  horizontal,  and  re- 


250  THE   ELEMEN7TS  OF  MECHANICS.  CHAP.  XXI. 

sumes  that  position  when  removed  from  it,  as  also  when  the 
steelyard  is  loaded,  and  the  Aveight  is  at  the  division  which 
ought  to  show  how  much  the  merchandise  weighs.  The  hor- 
izontal situation  in  this  steelyard,  as  well  as  in  the  others,  is 
known  by  means  of  the  tongue,  which  rises  vertically  above 
the  axis  of  suspension. 

3.  It  may  be  discovered  that  the  steelyard  is  deranged,  if, 
when  not  loaded,  the  beam  does  not  remain  horizontal. 

4.  The  advantage  of  a  great  and  a  small  side  (which  in  the 
other  augments  the  extent  of  their  power  of  weighing)  is  sup- 
plied by  a  very  simple  process,  which  accomplishes  the  same 
end  with  some  additional  advantages.     This  process  is  to  em- 
ploy on  the  same  division  different  weights.     The  numbers  of 
the  divisions  on  the  bar  point  out  the  degree  of  heaviness  ex- 
pressed by  the  corresponding  weights.     For  example,  when 
the   large   weight   of  the   large   steelyard    weighs    16   Ibs., 
each   division  it  passes  over  on  the  bar  is  equivalent  to  a 
pound ;   the  small  weight,  weighing  sixteen  times  less  than 
the  large  one,  will  represent  on  each  of  these  divisions  the 
sixteenth  part  of  a  pound,  or  one  ounce ;  and  the  opposite 
face  of  the  bar  is  marked  by  pounds  at  each  sixteenth  divis- 
ion.     In  this  construction,  therefore,  we   have  the   advan- 
tage of  being  able,  by  employing  both  weights  at  once,  to 
ascertain,  for  example,  almost  within  an  ounce,  the  weight  of 
500  pounds  of  merchandise.     It  will  be  sufficient  to  add 
what  is  indicated  by  the  small  weight  in  ounces,  to  that  of 
the  large  one  in  pounds,  after  an  equilibrium  has  been  obtain- 
ed by  the  position  of  the  two  weights,  viz.  the  large  one  placed 
at  the  next  pound  below  its  real  weight,  and  the  small  one 
at  the  division  which  determines  the  number  of  ounces  to  be 
added. 

5.  As  the  beam  is  graduated  only  on  one  edge,  it  may 
have  the  form  of  a  thin  bar,  which  renders  it  much  less 
susceptible  of  being  bent  by  the  action  of  the  weight,  and 
affords  room  for  making  the  figures  more  visible  on  both  the 
faces. 

6.  In  these  steelyards,  the  disposition  of  the  axes  is  not  only 
such  that  the  beam  represents  a  mathematical  lever  without 
weight,  but  in  the  principle  of  its  division,  the  interval  be- 
tween every  two  divisions  is  a  determined  and  aliquot  part  of 
the  distance  between  the  two  fixed  points  of  suspension  ;  and 
each  of  the  two  weights  employed  has  for  its  absolute  weight 
the  unity  of  the  weight  it  represents,  multiplied  by  the  num 


CHAP.  xxi.  c.  PAUL'S  STEELYARD.  251 

ber  of  the  divisions  contained  in  the  interval  between  the  two 
centres  of  motion. 

Thus,  supposing  the  arms  of  the  steelyard  divided  in  such 
a  manner  that  ten  divisions  are  exactly  contained  in  the  dis- 
tance between  the  two  constant  centres  of  motion,  a  weight 
to  express  the  pounds  on  each  division  of  the  beam,  must 
really  weigh  ten  pounds ;  that  to  point  out  the  ounces  on  the 
same  divisions  must  weigh  ten  ounces,  &,c. ;  so  that  the  same 
steelyard  may  be  adapted  to  any  system  of  measures  whatever, 
and  in  particular  to  the  decimal  system,  by  varying  the  ab- 
solute heaviness  of  the  weights,  and  their  relation  with  each 
other. 

But  to  trace  out,  in  a  few  words,  the  advantages  of  the  steel- 
yards constructed  by  C.  Paul  for  commercial  purposes,  we 
shall  only  observe, — 

1.  That  the  buyer  and  seller  are  certain  of  the  correctness 
of  the  instrument,  if  the  beam  remains  horizontal  when  it  is 
unloaded  and  in  its  usual  position.  2.  That  these  steelyards 
have  one  suspension  less  than  the  old  ones,  and  are  so  much 
more  simple.  3.  That  by  these  means  we  obtain,  with  the 
greatest  facility,  by  employing  two  weights,  the  exact  weight  of 
merchandise,  with  all  the  approximation  that  can  be  desired, 
and  even  with  a  greater  precision  than  that  given  by  common 
balances.  There  are  few  of  these  which,  when  loaded  with 
500  pounds  at  each  end,  give  decided  indication  of  an  ounce 
variation  ;  and  the  steelyards  of  C.  Paul  possess  that  advan- 
tage, and  cost  one  half  less  than  balances  of  equal  dominion. 
4.  In  the  last  place,  we  may  verify  at  pleasure  the  justness  of 
the  weights,  by  the  transposition  which  their  ratio  to  each 
other  will  permit ;  for  example,  by  observing  whether,  when 
the  weight  of  one  pound  is  brought  back  one  division,  and 
the  weight  of  one  ounce  carried  forward  sixteen  divisions,  the 
equilibrium  still  remains. 

It  is  on  this  simple  and  advantageous  principle  that  C.  Paul 
has  constructed  his  universal  steelyard.  It  serves  for  weigh- 
ing in  the  usual  manner,  and  according  to  any  system  of 
weights,  all  ponderable  bodies  to  the  precision  of  half  a  grain 
in  the  weight  of  a  hundred  ounces ;  that  is  to  say,  of  a  ten 
thousandth  part.  It  is  employed,  besides,  for  ascertaining  the 
specific  gravity  of  solids,  of  liquids,  and  of  air,  by  processes 
extremely  simple,  and  which  do  not  require  many  subdivisions 
in  the  weights. 

We  think  the  description  above  given  will  be  sufficiently 


252  THE  ELEMENTS  OF  MECHANICS.     CHAP.  XXI. 

intelligible  without  a  representation  of  this  instrument.  An 
account  of  its  application  to  the  determination  of  specific  grav- 
ities will  be  found  in  vol.  iii.  of  the  Philosophical  Magazine. 

The  Chinese  Steelyard. 

This  instrument  is  used  in  China  and  the  East  Indies  for 
weighing  gems,  precious  metals,  &>c.  The  beam  is  a  small 
rod  of  ivory,  about  a  foot  in  length.  Upon  this  are  three  lines 
of  divisions,  marked  by  fine  silver  studs,  all  beginning  from 
the  end  of  the  beam,  whence  the  first  is  extended  8  inches, 
the  second  6£,  and  the  third  8£.  The  first  is  European 
weight,  and  the  other  two  Chinese.  At  the  other  end  of  the 
beam  hangs  a  round  scale,  and  at  three  several  distances  from 
this  end  are  holes,  through  which  pass  so  many  fine  strings, 
serving  as  different  points  of  suspension.  The  first  distance 
makes  If  inches,  the  second  3£,  or  double  the  former,  and 
the  third  4£,  or  triple  the  same.  The  instrument,  when 
used,  is  held  by  one  of  the  strings,  and  a  sealed  weight  of 
about  l£  oz.  troy,  is  slid  upon  the  beam  until  an  equilibrium 
is  produced ;  the  weight  of  the  body  is  then  indicated  by 
the  graduated  scale  above-mentioned. 

The  Danish  Balance. 

The  Danish  balance  is  a  straight  bar  or  lerer  having  a 
heavy  weight  fixed  to  one  end,  and  a  hook  or  scale-pan  to  re- 
ceive the  substance,  the  weight  of  which  is  to  be  determined, 
suspended  from  the  other  end.  The  fulcrum  is  movable,  and 
is  made  to  slide  upon  the  bar,  till  the  beam  rests  in  a  horizon- 
tal position,  when  the  place  of  the  fulcrum  indicates  the  weight 
required.  In  order  to  construct  a  balance  of  this  kind,  let 
the  distance  of  the  centre  of  gravity  from  that  point  to  which 
the  substance  to  be  weighed  is  suspended  be  found  by  exper- 
iment, when  the  beam  is  unloaded.  Multiply  this  distance 
by  the  weight  of  the  whole  apparatus,  and  divide  the  product 
by  the  weight  of  the  apparatus  increased  by  the  weight  of  the 
body.  This  will  give  the  distance  from  the  point  of  suspen- 
sion, at  which  the  fulcrum  being  placed,  the  whole  will  be  in 
equilibrio :  for  example,  supposing  the  distance  of  the  centre 
of  gravity  from  the  point  of  suspension  to  be  10  inches,  and 
the  weight  of  the  whole  apparatus  to  be  ten  pounds ;  sup- 
pose, also,  it  were  required  to  mark  the  divisions  which 


CHAP.  XXI.   BENT  LEVER  BALANCE  -  BRADY'S  BALANCE.   253 

should  indicate  weights  of  one,  two,  or  three  pounds,  &,c. 
First,  for  the  place  of  the  division  indicating  one  pound  we 


have  =  =  9i*r  inches>  the  Place  of  the 


marking  one  pound.  For  two  pounds  we  have 
inches,  the  place  of  the  division  indicating  two  pounds  ;  and 
for  three  pounds  10?3  =  7^  inches  for  the  place  of  the  divis- 
ions indicating  three  pounds,  and  so  on. 

This  balance  is  subject  to  the  inconvenience  of  the  divis- 
ions becoming  much  shorter  as  the  weight  increases.  The 
distance  between  the  divisions  indicating  one  and  two  pounds 
being,  in  the  example  we  have  given,  about  seven  tenths  of 
an  inch,  whilst  that  between  20  and  21  pounds  is  only  one 
tenth  of  an  inch  ;  consequently,  a  very  small  error  in  the 
place  of  the  divisions  indicating  the  larger  weights  would 
occasion  very  inaccurate  results.  The  Danish  balance  is 
represented  ztjig.  194. 

The  Bent  Lever  Balance. 

This  instrument  is  represented  at  Jig.  195.  The  weight  at 
C  is  fixed  at  the  end  of  the  bent  lever  ABC,  which  is  sup- 
ported by  its  axis  B  on  the  pillar  I  H.  A  scale-pan  E  is  sus- 
pended from  the  other  end  of  the  lever  at  A.  Through  the 
centre  of  motion  B  draw  the  horizontal  line  K  B  G,  upon 
which,  from  A  and  C,  let  fall  the  perpendiculars  A  K  and  C 
D.  Then,  if  B  K  and  B  D  are  reciprocally  proportional  to 
the  weights  at  A  and  C,  they  will  be  in  equilibrio,  but  if  not, 
the  weight  C  will  move  upwards  or  downwards  along  the  arc 
F  G  till  that  ratio  is  obtained.  If  the  lever  be  so  bent  that 
when  A  coincides  with  the  line  G  K,  C  coincides  with  the 
vertical  B  H,  then  as  C  moves  from  F  to  G,  its  momentum 
will  increase  while  that  of  the  weight  in  the  scale-pan  E  will 
decrease.  Hence  the  weight  in  E,  corresponding  to  different 
positions  of  the  balance,  may  be  expressed  on  the  graduated 
arc  F  G. 

Brady1  s  Balance,  or  Weighing  Apparatus. 

This  partakes  of  the  properties  both  of  the  bent  level 
balance  and  of  the  steelyard.  It  is  represented  zijig.  196 
A  B  C  is  a  frame  of  cast  iron  having  a  great  part  of  its  weight 

22 


254  THE  ELEMENTS  OF  MECHANICS.      CHAP.  XXI. 

towards  A.  F  is  a  fulcrum,  and  E  a  movable  suspender, 
having  a  scale  and  hook  at  its  lower  extremity.  E  K  G  are 
three  distinct  places,  to  which  the  suspender  E  may  be  appli- 
ed, and  to  which  belong  respectively  the  three  graduated 
scales  of  division  expressing  weights,  fG,  c  d,  and  a  b. 
When  the  scale  and  suspender  are  applied  at  G,  the  appara- 
tus is  in  equilibrio,  with  the  edge  A  B  horizontal,  and  the 
suspender  cuts  the  zero  on  the  scale  a  b.  Now,  any  sub- 
stance, the  weight  of  which  is  to  be  ascertained,  being  put 
into  the  scale,  the  whole  apparatus  turns  about  F,  and  the 
part  towards  B  descends  till  the  equilibrium  is  again  estab- 
lished, when  the  weight  of  the  body  is  read  off  from  the  scale 
a  b,  which  registers  to  ounces  and  extends  to  two  pounds. 
If  the  weight  of  the  body  exceed  two  pounds,  and  be  less 
than  eleven  pounds,  the  suspender  is  placed  at  K ;  and  when 
the  scale  is  empty,  the  number  2  is  found  to  the  right  of  the  in- 
dex of  the  suspender.  If  now  weights  exceeding  two  pounds 
be  placed  in  the  scale,  the  whole  again  turns  about  F,  and 
the  weight  of  the  body  is  shown  on  the  graduated  arc  c  d, 
which  extends  to  eleven  pounds,  and  registers  to  every  two 
ounces. 

If  the  weight  of  the  body  exceed  eleven  pounds,  the  sus- 
pender is  hung  on  at  E,  and  the  weights  are  ascertained  in 
the  same  manner  on  the  scale  fC  to  thirty  pounds,  the  sub- 
divisions being  on  this  scale  quarters  of  pounds.  The  same 
principles  would  obviously  apply  to  weights  greater  or  less 
than  the  above.  To  prevent  mistake,  the  three  points  of  sup- 
port G,  K,  E,  are  numbered  1,  2,  3;  and  the  corresponding 
arcs  are  respectively  numbered  in  the  same  manner.  When 
the  hook  is  used  instead  of  the  scale,  the  latter  is  turned  up- 
wards, there  being  a  joint  at  m  for  that  purpose. 

The  Weighing  Machine  for  Turnpike  Roads. 

This  machine  is  for  the  purpose  of  ascertaining  the  weight 
of  heavy  bodies,  such  as  wheel  carriages.  It  consists  of  a 
wooden  platform  placed  over  a  pit  made  in  the  line  of  the  road, 
and  which  contains  the  machinery.  The  pit  is  walled  within- 
side,  and  the  platform  is  fitted  to  the  walls  of  the  pit,  but  with- 
out touching  them,  and  it  is  therefore  at  liberty  to  move  freely 
up  and  down.  The  platform  is  supported  by  levers  placed 
beneath  it,  and  is  exactly  level  with  the  surface  of  the  road, 
so  that  a  carriage  is  easily  drawn  on  it,  the  wheels  being  upon 


CHAP.  XXI.  WEIGHING    MACHINE.  255 

the  platform  whilst  the  horses  are  upon  the  solid  ground  beyond 
it.  The  construction  of  this  machine  will  be  readily  under- 
stood by  reference  to  Jig.  197.,  in  which  the  platform  is  sup- 
posed to  be  transparent,  so  as  to  allow  of  the  levers  being 
seen  below  it. 

A,  B,  C,  D,  represent  four  levers  tending  towards  the  cen- 
tre of  the  platform,  and  each  movable  on  its  fulcrum  at  A,  B, 
C,  D ;  the  fulcrum  of  each  rests  upon  a  piece  securely  fixed 
in  the  corner  of  the  pit.  The  platform  is  supported  upon  the 
cross  pins  a,  6,  c,  d,  by  means  of  pieces  of  iron  which  pro- 
ject from  it  near  its  corners,  and  which  are  represented  in  the 
plate  by  the  short  dark  lines  crossing  the  pins  a,  6,  c,  d.  The 
four  levers  are  connected  under  the  centre  of  the  platform, 
but  not  so  as  to  prevent  their  free  motion,  and  are  supported 
by  a  long  lever  at  the  point  F,  the  fulcrum  of  which  rests  upon 
a  piece  of  masonry  at  E  :  the  end  of  this  last  lever  passes  be- 
low the  surface  of  the  road  into  the  turnpike  house,  and  is 
there  attached  to  one  arm  of  a  balance,  or,  as  in  Salmon's 
patent  weighing  machine,  to  a  strap  passing  round  a  cylinder 
which  winds  up  a  small  weight  round  a  spiral,  and  indicates, 
by  means  of  an  index,  the  weight  placed  upon  the  platform. 

Suppose  the  distance  from  A  to  F  to  be  ten  times  as  great 
as  that  from  A  to  a,  then  a  force  of  one  pound  applied  be- 
neath F  would  balance  ten  pounds  applied  at  a,  or  upon  the 
platform.  Again :  let  the  distance  from  E  to  G  be  also  ten 
times  greater  than  the  distance  from  the  fulcrum  E  to  F ; 
then  a  force  of  one  pound  applied  to  raise  up  the  end  of  the 
lever  G  would  counterpoise  a  weight  often  pounds  placed  up- 
on F.  Now,  as  we  gain  ten  times  the  power  by  the  first  levers, 
and  ten  times  more  by  the  lever  E  G,  it  follows,  that  a  force  of 
one  pound  tending  to  elevate  G,  would  balance  100  Ibs.  placed 
on  the  platform  ;  so  that  if  the  end  of  the  lever  G  be  attached 
to  one  arm  of  a  balance,  a  weight  of  10  Ibs.  placed  in  a  scale 
suspended  from  the  other  arm,  will  express  the  value  of  1000 
Ibs.  placed  upon  the  platform.  The  levers  are  counterpoised, 
when  the  platform  is  not  loaded,by  a  weight  H  applied  to  the  end 
of  the  last  lever,  continued  beyond  the  fulcrum  for  that  purpose. 

Of  Instruments  for  weighing  by  Means  of  a  Spring. 

The  spring  is  well  adapted  to  the  construction  of  a  weighing 
machine,  from  the  property  it  possesses  of  yielding  in  propor- 
tion to  the  force  impressed,  and  consequently  giving  a  scale 


1256  THE  ELEMENTS  OF  MECHANICS.  CHAP.  XXI. 

of  equal  parts  for  equal  additions  of  weight.  It  is  liable, 
however,  to  suffer  injury,  unless  the  steel  of  which  it  is  com- 
posed be  very  well  tempered,  from  a  want  of  perfect  elasticity, 
and,  consequently,  from  not  returning  to  its  original  place 
after  it  has  been  forcibly  compressed.  This,  however,  must 
be  considered  to  arise,  in  a  great  measure,  from  imperfection 
of  workmanship,  or  of  the  material  employed,  or  to  its  hav- 
ing been  subjected  to  too  great  a  force. 

The  Spring  Steelyard. 

The  little  instrument  known  by  this  name  is  in  very  gen- 
eral use,  and  is  particularly  convenient  where  great  accuracy 
is  not  necessary,  as  a  spring,  which  will  ascertain  weights 
from  one  pound  to  fifty,  is  contained  in  a  cylinder  only  4 
inches  long  and  £  inch  diameter. 

This  instrument  is  represented  at  Jig.  198.  It  consists  of 
a  tube  of  iron,  of  the  dimensions  just  stated,  closed  at  the 
bottom,  to  which  is  attached  an  iron  hook,  for  supporting  the 
substance  to  be  weighed ;  a  rod  of  iron  a  b,  four  tenths  of 
an  inch  wide  and  one  tenth  thick,  is  firmly  fixed  in  the  cir- 
cular plate  c  d,  which  slides  smoothly  in  the  iron  tube.* 

A  strong  steel  spring  is  also  fastened  to  this  plate,  and 
passed  round  the  rod  a  b  without  touching  it,  and  without 
coming  in  contact  with  the  interior  of  the  cylindrical  tube. 
The  tube  is  closed  at  the  top  by  a  circular  piece  of  iron 
through  which  the  piece  a  b  passes. 

Upon  the  face  of  a  b  the  weight  is  expressed  by  divisions, 
each  of  which  indicates  one  pound,  and  five  of  such  divisions 
in  the  instrument  now  before  us  occupy  two  tenths  of  an  inch. 
The  divisions,  notwithstanding,  are  of  sufficient  size  to  ena- 
ble them  to  be  subdivided  by  the  eye. 

To  use  this  instrument,  the  substance  to  be  weighed  is 
suspended  by  the  hook,  the  instrument  being  held  by  a  ring 
passing  through  the  rod  at  the  other  end.  The  spring  then 
suffers  a  compression  proportionate  to  the  weight,  and  the 
number  of  pounds  is  indicated  by  the  division  on  the  rod 
which  is  cut  by  the  top  of  the  cylindrical  tube. 

Salter's  improved  Spring  Balance. 

A  very  neat  form  of  the  instrument  last  described  has 
been  recently  brought  before  the  public  by  Mr.  Salter,  under 


CHAP.    XXI.  DYNAMOMETER.  257 

the  name  of  the  Improved  Spring  Balance.  It  is  represented 
•Aijig.  199.  The  spring  is  contained  in  the  upper  half  of  a 
cylinder  behind  the  brass  plate  forming  the  face  of  the  in- 
strument ;  and  the  rod  is  fixed  to  the  lower  extremity  of  the 
spring,  which  is  consequently  extended,  instead  of  being  com- 
pressed, by  the  application  of  the  weight.  The  divisions, 
each  indicating  half  a  pound,  are  engraved  upon  the  face  of 
the  brass  plate,  and  are  pointed  out  by  an  index  attached  to 
the  rod. 

Marriott's  Patent  Dial  Weighing  Machine. 

The  exterior  of  this  instrument  is  represented  vtjig.  200., 
and  the  interior  at  Jig.  201.  A  B  C  is  a  shallow  brass  box, 
having  a  solid  piece  as  represented  at  A,  to  which  the  spring 
D  E  F  is  firmly  fixed  by  a  nut  at  D.  The  other  end  of  the 
spring  at  F  is  pinned  to  the  brass  piece  G  H,  to  the  part  of 
which  at  G  is  also  fixed  the  iron  racked  plate  I.  A  screw  L 
serves  as  a  stop  to  keep  this  rack  in  its  place.  The  teeth  of 
the  rack  fit  into  those  of  the  pinion  M,  the  axis  of  which 
passes  through  the  centre  of  the  dial-plate,  and  carries  ari 
index  which  points  out  the  weight.  The  brass  piece  G  H  is 
merely  a  plate  where  it  passes  over  the  spring,  and  the  tail 
piece  H,  to  which  the  weight  is  suspended,  passes  through 
an  opening  in  the  side  of  the  box. 

Of  the  Dynamometer. 

This  is  an  important  instrument  in  mechanics,  calculated 
to  measure  the  muscular  strength  exerted  by  men  and  ani- 
mals. It  consists  essentially  of  a  spring  steelyard,  such  as 
that  we  first  described.  This  is  sometimes  employed  alone, 
and  sometimes  in  combination  with  various  levers,  which 
allow  of  the  spring  being  made  more  delicate,  and  conse- 
quently increase  the  extent  of  the  divisions  indicating  the 
weight. 

The  first  instrument  of  this  kind  appears  to  have  been 
invented  by  Mr.  Graham,  but  it  was  too  bulky  and  incon- 
venient for  use.  M.  le  Roy  made  one  of  a  more  simple 
construction.  It  consisted  of  a  metal  tube,  about  a  foot  long, 
placed  vertically  upon  a  stand,  and  containing  in  the  inside 
a  spiral  spring,  having  above  it  a  graduated  rod  terminating 
in  a  globe.  This  rod  entered  the  tube  more  or  less  in  propor- 
22* 


258  THE  ELEMENTS  OF  MECHANICS.  CHAP.  XXI. 

tion  to  the  force  applied  to  the  globe,  and  the  divisions 
indicated  the  quantity  of  this  force.  Therefore,  when  a 
man  pressed  upon  the  globe  with  all  his  strength,  the  divis- 
ions upon  the  rod  showed  the  number  of  pounds  weight  to 
which  it  was  equal. 

An  instrument  of  this  kind  for  determining  the  force  of  a 
blow  struck  by  a  man  with  his  fist  was  lately  exhibited  at  the 
National  Repository.  It  was  fixed  to  a  wall,  from  which  it 
projected  horizontally.  In  place  of  the  globe  there  was  a 
cushion  to  receive  the  blow,  and  as  the  suddenness  with 
which  the  spring  returned  rendered  it  impossible  to  read  the 
division  upon  the  rod,  another  rod,  similarly  divided,  was 
forced  in  by  the  plate  forming  the  basis  of  the  cushion,  and 
remained  stationary  when  the  spring  returned.  The  com- 
mon spring  steelyard,  however,  which  we  first  described,  is 
in  principle  'the  same  as  M.  le  Roy's  dynamometer,  and  is 
much  more  conveniently  constructed  for  the  purpose  we  are 
considering.  The  ring  at  one  end  may  be  fixed  to  an  im- 
movable object,  and  the  hook  at  the  other  attached  to  a  man, 
or  to  an  animal,  and  the  extent  to  which  the  graduated  rod 
is  drawn  out  of  the  cylinder  shows  at  once  the  force  which 
is  applied.  Though  this  is  perhaps  the  best,  and  certainly 
the  most  simple  dynamometer,  others  have  been  contrived, 
which  are,  however,  but  modifications  of  the  spring  steelyard. 
One  of  these  is  represented  at  fig.  202.  The  spiral  spring 
acts  in  the  manner  before  described,  but  its  divisions  are  in- 
creased in  size,  and  therefore  rendered  more  perceptible  by 
means  of  a  rack  fixed  to  the  plate,  acting  against  the  spiral 
spring,  the  teeth  of  which  move  a  pinion  upon  which  the 
arm  I  is  fixed,  pointing  to  the  graduated  arc  K. 

Another  dynamometer  has  been  invented  by  Mr.  Salmon  ; 
it  is  represented  atj^.  203.,  and  is  a  combination  of  levers 
with  the  spring.  By  means  of  these  levers,  a  much  more 
delicate  spring,  and  which  is  therefore  more  sensible,  may  be 
employed  than  in  the  dynamometer  last  described. 

The  manner  in  which  these  levers  and  springs  act  will  be 
readily  understood  by  an  inspection  of  the  figure.  Like  the 
weighing  machine  for  carriages,  the  fulcrum  of  each  lever  is 
at  one  end,  and  the  force  is  diminished,  in  passing  to  the 
spring,  in  the  ratio  of  the  length  of  its  arms.  The  spring 
moves  a  pinion  by  means  of  a  rack,  upon  which  pinion  a 
hand  is  placed,  indicating  by  divisions  upon  a  circular  dial- 
plate  the  amount  of  the  force  employed. 


CHAP.  XXI.      COMPENSATION  PENDULUMS.  259 

The  spring  used  in  this  machine  is  calculated  to  weigh 
only  about  50  Ibs.  instead  of  about  5  cwt.,  as  in  the  last  de- 
scribed ;  but  by  means  of  the  levers  which  intervene  between 
it  and  the  force  applied,  it  will  serve  to  estimate  a  force  equal 
to  6  cwt,  and  might  obviously  be  made  to  go  to  a  much 
greater  extent,  by  varying  the  ratio  of  the  length  of  the  arms 
of  the  levers. 

ON    COMPENSATION    PENDULUMS. 

(336.)  It  is  said  of  Galileo,  that,  when  very  young,  he  ob- 
served a  lamp  suspended  from  the  roof  of  a  church  at  Pisa, 
swinging  backwards  and  forwards  with  a  pendulous  motion. 
This,  if  it  had  been  remarked  at  all  by  an  uneducated 
mind,  would,  most  probably,  have  been  passed  by  as  a  com- 
mon occurrence,  unworthy  of  the  slightest  notice  ;  but  to  the 
mind  imbued  with  science  no  incident  is  insignificant ;  and 
a  circumstance  apparently  the  most  trivial,  when  subjected 
to  the  giant  force  of  expanded  intellect,  may  become  of  im- 
mense importance  to  the  improvement  and  to  the  well-being 
of  man.  The  fall  of  an  apple,  it  is  said,  suggested  to 
Newton  the  theory  of  gravitation,  and  his  powerful  inind 
speedily  extended  to  all  creation  that  great  law  which  brings 
an  apple  to  the  ground.  The  swinging  of  a  lamp  in  a  church 
at  Pisa,  viewed  by  the  piercing  intellect  of  Galileo,  gave  rise 
to  an  instrument  which  affords  the  most  perfect  measure  of 
time,  which  serves  to  determine  the  figure  of  the  earth,  and 
which  is  inseparably  connected  with  all  the  refinements  of 
modern  astronomy. 

The  properties  of  the  pendulum,  and  the  manner  in  which 
it  serves  to  measure  time,  have  been  fully  explained  in  Chap- 
t3r  XI. :  and  if  a  substance  could  be  found  not  susceptible 
of  any  change  in  its  dimensions  from  a  change  of  tempera- 
ture, nothing  more  would  be  necessary,  as  the  centre  of 
oscillation  would  always  remain  at  the  same  distance  from 
the  point  of  suspension.  As  every  known  substance,  how- 
ever, expands  with  heat,  and  contracts  with  cold,  the  length 
of  the  pendulum  will  vary  with  every  alteration  of  tempera- 
ture, and  thus  the  time  of  its  vibration  will  suffer  a  corre- 
sponding chancre.  The  effect  of  a  difference  of  temperature 
of  25°,  or  that  which  usually  occurs  between  winter  and 
summer,  would  occasion  a  clock  furnished  with  a  pendulum 
having  an  iron  rod  to  gain  or  lose  six  seconds  in  twenty-four 
hours. 


260  THE    ELEMENTS    OF    MECHANICS.  CHAP.  XXI. 

It  became,  then,  highly  important  to  discover  some  means 
of  counteracting  this  variation  to  which  the  length  of  the 
pendulum  was  liable,  or,  in  other  words,  to  devise  a  method 
by  which  the  centre  of  oscillation  should,  under  every  change 
of  temperature,  remain  at  the  same  distance  from  the  point 
of  suspension  :  happily,  the  difference  in  the  rate  of  expansion 
of  different  metals  presented  a  ready  means  of  effecting  this. 

Graham,  in  the  year  1715,  made  several  experiments  to 
ascertain  the  relative  expansions  of  various  metals,  with  a 
view  of  availing  himself  of  the  difference  of  the  expansions 
of  two  or  more  of  them  when  opposed  to  each  other,  to  con- 
struct a  compensating  pendulum.  But  the  difference  he 
found  was  so  small,  that  he  gave  up  all  hope  of  being  able  to 
accomplish  his  object  in  that  way.  Knowing,  however,  that 
mercury  was  much  more  affected  by  a  given  change  of  tem- 
perature than  any  other  substance,  he  saw  that  if  the  mercu- 
ry could  be  made  to  ascend  while  the  rod  of  the  pendulum 
became  longer,  and  vice,  vcrzti,  the  centre  of  oscillation  might 
always  be  kept  at  the  same  distance  from  the.  point  of  sus- 
pension. This  idea  happily  gave  birth  to  the  mercurial 
pendulum,  which  is  now  in  very  general  use. 

In  the  mean  time,  Graham's  suggestion  excited  the  inge- 
nuity of  Harrison,  originally  a  carpenter  at  Barton  in  Lin- 
colnshire, who,  in  1720,  produced  a  pendulum  formed  of 
parallel  brass  and  steel  rods,  known  by  the  name  of  the 
gridiron  pendulum. 

In. the  mercurial  pendulum,  the  bob  or  weight  is  the  mate- 
rial affording  the  compensation  ;  but  in  the  gridiron  pendu- 
lum, the  object  is  attained  by  the  greater  expansion  of  the 
brass  rods,  which  raise  the  bob  upwards  towards  the  point  of 
suspension  as  much  as  the  steel  rods  elongate  downwards. 

In  the  present  article,  we  shall  describe  such  compensation 
pendulums  as  appear  to  us  likely  to  answer  best  in  practice ; 
and  we  trust  we  shall  be  able  to  simplify  the  subject  so  as  to 
render  a  knowledge  of  mathematics  in  the  construction  of 
this  important  instrument  unnecessary. 

The  following  table  contains  the  linear  expansion  of  vari- 
ous substances  in  parts  of  their  length,  occasioned  by  a 
change  of  temperature  amounting  to  one  degree.  We  have 
taken  the  liberty  of  extracting  it  from  a  very  valuable  paper 
by  F.  Bailey,  Esq.,  on  the  mercurial  compensation  pendulum, 
published  in  the  Memoirs  of  the  Astronomical  Society  of 
London  for  1824. 


CHAP.  XXI. 


COMPENSATION  PENDULUMS. 


261 


TABLE  I. 

Linear  Expansion  of  Various  Substances  for  one  Degree  of 
Fahrenheit's  Thermometer. 


Substances. 

Expansions. 

Authors. 

White  Deal,    .... 
English  Flint  Glass, 
Iron  (cast),    

i 
j 

•0000022685 
•0000028444 
•0000047887 
•0000061700 

Captain  Kater. 
Dr.  Struve. 
Dulong  and  Petit. 
General  Roy. 

Iron  (wire),  .  .  .  .  . 

1 

•0000065668 
•0000068613 

Dulong  and  Petit. 
Lavoisier  and  L. 

•0000069844 

Hasslar. 

Steel  (rod),  

•0000063596 

General  Roy. 

Brass,   . 

.0000104400 

C  Commiss.  of  Weights 
<  and  Measures  —  mean 

•0000159259 

<J  of  several  experiments. 
Smeatofu 

•0000163426 

Ditto. 

Zinc  (hammered),   . 
Mercury  in  bulk,  .  . 

•0000172685 
•00010010 

Ditto. 
Dulong  and  Petit. 

From  this  table  it  is  easy  to  determine  the  length  of  a  rod 
of  any  substance,  the  expansion  of  which  shall  be  equal  to 
that  of  a  rod  of  given  length,  of  any  other  substance. 

The  lengths  of  such  rods  will  be  inversely  proportionate  to 
their  expansions.  If,  therefore,  we  divide  the  lesser  expan- 
sion by  the  greater  (supposing  the  rod  the  length  of  which  is 
given  to  be  made  of  the  lesser  expansible  material),  and  mul- 
tiply the  given  length  by  this  quotient,  we  shall  have  the 
required  length  of  a  rod,  the  expansion  of  which  will  be 
equal  to  that  of  the  rod  given.  For  example  : — The  ex- 
pansion of  a  rod  of  steel  being,  from  the  above  table, 
•0000063596,  and  that  of  brass,  -0000104400;  if  it  were 
required  to  determine  the  length  of  a  rod  of  brass  which 
should  expand  as  much  as  a  rod  of  steel  of  39  inches  in 

length,  we  have  .OOU0104^  =  '6091,  which,  multiplied  by  39, 
gives  23-75  inches  for  the  length  of  brass  required. 

We  shall  here,  in  order  to  facilitate  calculation,  give  the 


262  THE  ELEMENTS  OF  MECHANICS.     CHAP.  XXI. 

ratio  of  the  lengths  of  such  substances  as  may  be  employed 
in  the  construction  of  compensation  pendulums. 


TABLE  II. 


Steel  rod  and  brass  compensation,  as  1, 

Iron  wire  rod  and  lead  compensation, -4308 

Steel  rod  and  lead  compensation, '3993 

Iron  wire  rod  and  zinc  compensation, '3973 

Steel  rod  and  zinc  compensation, *3682 

Glass  rod  and  lead  compensation, *3007 

Glass  rod  and  zinc  compensation, -2773 

Deal  rod  and  lead  compensation, -1427 

Deal  rod  and  zinc  compensation, -1313 

Steel  rod  and  mercury  in  a  steel  cylinder, -0728 

Steel  rod  and  mercury  in  a  glass  cylinder, -0703 

Glass  rod  and  mercury  in  a  glass  cylinder, '0529 


It  is  evident  that  in  this  table  the  decimals  express  the 
length  of  a  roc!  of  the  compensating  material,  the  expansion 
of  which  is  equal  to  that  of  a  pendulum  rod  whose  length  is 
unity. 

As  we  are  not  aware  of  the  existence  of  any  work  which 
contains  instructions  that  might  enable  an  artist  or  an  ama- 
teur to  make  a  compensation  pendulum,  we  shall  endeavor  to 
give  such  detailed  information  as  may  free  the  subject  from 
every  difficulty. 

The  pendulum  of  a  clock  is  generally  suspended  by  a 
spring,  fixed  to  its  upper  extremity,  and  passing  through  a 
slit  made  in  a  piece  which  is  called  the  cock  of  the  pendu- 
lum. The  point  of  suspension  is,  therefore,  that  part  of  the 
spring  which  meets  the  lower  surface  of  the  cock.  Now  the 
istance  of  the  centre  of  oscillation  of  the  pendulum  from 
his  point  may  be  varied  in  two  ways  ;  the  one  by  drawing 
up  the  spring  through  this  slit,  and  the  other  by  raising  the 
bob  of  the  pendulum.  Either  of  these  methods  may  be 
practised  in  the  compensation  pendulum,  but  the  former  is 
subject  to  objections  from  which  the  latter  is  exempt. 

Suppose  it  were  required  to  compensate  a  pendulum  of  39 
inches  in  length,  of  steel,  by  means  of  the  expansion  of  a 
brass  rod.  Here,  referring  to  -Jf.g.  204.,  we  have  S  C  39 
inches  (which  is  to  remain  constant)  of  steel ;  the  pendulum 


CHAP.  XXI.  COMPENSATION    PENDULUMS.  263 

spring,  passing  through  the  cock  at  S,  is  attached  to  another 
rod  of  steel,  which  is  fixed  to  the  cross  piece  R  A  at  A. 
The  other  end  of  the  cross  piece  at  R  is  fastened  to  a  brass 
rod,  the  lower  extremity  of  which  is  fixed  to  the  cock  of  the 
pendulum  at  B.  Now  the  brass  rod  B  R  must  expand  up- 
wards, as  much  as  the  steel  rod  A  C  expands  downwards ; 
and  the  length  of  the  brass  must  be  such  as  to  effect  this, 
leaving  39  inches  of  the  steel  rod  below  the  cock  of  the  pen- 
dulum. 

Let  us  first  try  80  inches  of  steel.  Multiplying  this  by 
•6091,  we  have  48*73  inches  for  the  length  of  brass,  which 
compensates  80  inches  of  steel.  But  as  48-73  inches  of  the 
steel,  equal  in  length  to  the  brass,  would  in  this  case  be 
above  the  cock  of  the  pendulum,  it  would  leave  only  31-27 
inches  below  it,  instead  of  39  inches. 

Let  us  now  try  100  inches  of  steel.  This,  multiplied  as 
before  by  -6091,  gives  60-91  inches,  according  to  the  expan- 
sions which  we  have  used,  for  the  length  of  the  brass  rod, 
and  leaves  39-09  inches  below  the  cock  of  the  pendulum, 
which  is  sufficiently  near  for  our  present  purpose. 

From  what  has  been  said,  we  may  perceive  that  the  total 
length  of  the  material  of  which  the  pendulum  rod  is  com- 
posed must  be  always  equal  to  the  length  of  the  pendulum 
added  to  the  length  of  the  compensation. 

In  this  instance  we  have  effected  our  object,  by  drawing 
the  pendulum-spring  through  the  slit ;  but  we  will  now  show 
how  the  same  thing  may  be  done  by  moving  the  bob  of  the 
pendulum.  At  Jig.  205.,  let  S  C,  as  before,  be  equal  to  39 
inches.  Let  the  steel  rod  S  D  turn  off  at  right  angles  at  D, 
and  let  a  rod  of  brass  B  R,  of  61  inches  in  length,  ascend 
perpendicularly  from  this  cross  piece  to  R.  To  the  upper 
part  of  the  brass  rod  fix  another  cross  piece  R  A,  and  from 
the  extremity  A  let  a  steel  rod  descend  to  E,  bending  it  as  in 
the  figure  till  it  reaches  C.  Now  the  total  length  of  the 
pieces  of  steel  expanding  downwards  is  equal  to  S  D,  D  F, 
and  F  C  (amounting  together  to  39  inches),  to  which  must 
be  added  a  length  of  steel  equal  to  that  of  the  brass  rod  B  R, 
(61  inches),  making  together  100  inches  of  steel,  as  before, 
the  expansion  of  which  downwards  is  compensated  by  that 
of  the  brass  rod,  of  61  inches  in  length,  expanding  upwards. 

This  form,  however,  is  evidently  inconvenient,  from  the 
great  length  of  brass  and  steel  which  is  carried  above  the 
cock  of  the  pendulum ;  but  it  is  the  same  thing  whether  the 


204  THE    ELEMENTS    OF    MECHANICS.  CHAP.    XXI. 

brass  and  steel  be  each  in  one  piece,  or  divided  into  several, 
provided  the  pieces  of  steel  be  all  so  arranged  as  to  expand 
downwards,  and  those  of  brass  upwards.  Thus,  ztjig.  206., 
the  portions  of  steel  expanding  downwards  are  together  equal, 
as  before,  to  100  inches,  and  the  two  brass  pieces  expanding 
upwards  are  together  equal  to  61  inches ;  so  that,  in  fact, 
the  two  last  forms  of  compensation  which  we  have  described 
differ  in  no  respect  from  each  other  in  principle,  but  only  in 
the  arrangement  of  the  materials.  The  last  is  the  half  of 
the  gridiron  pendulum,  the  remaining  bars  being  merely  du- 
plicates of  those  we  have  described,  and  serving  no  other 
purpose  but  to  form  a  secure  frame- work. 

Harrison's  Gridiron  Pendulum. 

After  what  has  been  said,  little  more  is  necessary  than  to 
give  a  representation  of  this  pendulum.  This  is  done  at 
jig.  207.,  in  which  the  darker  lines  represent  the  steel  rods, 
and  the  lighter  those  of  brass.  The  central  rod  is  fixed  at 
its  lower  extremity  to  the  middle  of  the  third  cross  piece  from 
the  bottom,  and  passes  freely  through  holes  in  the  cross 
pieces  which  are  above,  whilst  the  other  rods  are  secured 
near  their  extremities  to  the  cross  pieces  by  pins  passing 
through  them.  In  order  to  render  the  whole  more  secure, 
the  bars  pass  freely  through  holes  made  in  two  other  cross 
pieces,  the  extremities  of  which  are  fixed  to  the  exterior  steel 
wires.  As  different  kinds  of  the  same  metal  vary  in  their 
rate  of  expansion,  the  pendulum  when  finished  may  be  found 
upon  trial  to  be  not  duly  compensated.  In  this  case,  one  or 
more  of  the  cross  pieces  is  shifted  higher  or  lower  upon  the 
bars,  and  secured  by  pins  passed  through  fresh  holes. 

Troughton's  Tubular  Pendulum. 

This  is  an  admirable  modification  of  Harrison's  gridiron 
pendulum.  It  is  represented  atj^1.  208.,  where  it  may  be 
seen  that  it  has  the  appearance  of  a  simple  pendulum,  as  the 
whole  compensation  is  concealed  within  a  tube  six  tenths  of 
an  inch  in  diameter. 

A  steel  wire,  about  one  tenth  of  an  inch  in  diameter,  is 
fixed  in  the  usual  manner  to  the  sprihg  by  which  the  pendu- 
lum is  suspended.  This  wire  passes  to  the  bottom  of  an 
interior  brass  tube,  in  the  centre  of  which  it  is  firmly  screwed 


CHAP.  xxi.  TROUGH-TON'S  TUBULAR  PENDULUM.      265 

The  top  of  this  tube  is  closed,  the  steel  rod  passing  freely 
through  a  hole  in  the  centre.  Into  the  top  of  this  interior 
tube  two  steel  wires,  of  one  tenth  of  an  inch  in  diameter,  are 
screwed  into  holes  made  in  that  diameter,  which  is  at  right 
angles  to  the  motion  of  the  pendulum.  These  wires  pass 
down  the  tabe  without  touching  either  it  or  the  central  rod, 
through  holes  made  in  the  piece  which  closes  the  bottom  of 
the  interior  tube.  The  lower  extremities  of  these  wires, 
which  project  a  little  beyond  the  inner  tube,  are  securely 
fixed  in  a  piece  which  closes  the  bottom  of  an  exterior  brass 
tube,  which  is  of  such  a  diameter  as  just  to  allow. the  interior 
tube  to  pass  freely  through  it,  and  of  a  sufficient  length  to 
extend  a  little  above  it.  The  top  of  the  exterior  tube  is 
closed  like  that  of  the  interior,  having  also  a  hole  in  its  cen- 
tre, to  allow  the  first  steel  rod  to  pass  freely  through  it.  Into 
the  top  of  the  exterior  tube,  in  that  diameter  which  coincides 
with  the  motion  of  the  pendulum,  a  second  pair  of  steel  wires 
of  the  same  diameter  as  the  former  are  screwed,  their  dis- 
tance from  the  central  rod  being  equal  to  the  distance  of 
each  from  the  first  pair.  They  consequently  pass  down  with- 
in the  interior  tube,  and  through  holes  made  in  the  pieces 
closing  the  lower  ends  of  both  the  interior  and  exterior  tubes. 
The  lower  ends  of  these  wires  are  fastened  to  a  short  cylin- 
drical piece  of  brass  of  the  same  diameter  as  the  exterior 
tube,  to  which  the  bob  is  suspended  by  its  centre. 

Fig,  209.  is  a  full  sized  section  of  the  rod ;  the  three  con- 
centric circles  represent  the  two  tubes,  and  the  rectangular 
position  of  the  two  pair  of  wires  round  the  middle  one  is 
shown  by  the  five  small  circles. 

Fig.  210,  is  the  part  which  closes  the  upper  end  of  the 
interior  tube.  The  two  small  circles  are  the  two  wires  which 
proceed  from  it,  and  the  three  large  circles  show  the  holes 
through  which  the  middle  wire  and  the  other  pair  of  wires 
pass. 

Fig.  211.  is  the  bottom  of  the  interior  tube.  The  small 
circle  in  the  centre  is  where  the  central  rod  is  fastened  to  it, 
the  others  the  holes  for  the  other  four  wires  to  pass  through. 

Fig,  212.  is  the  part  which  closes  the  top  of  the  external 
tube.  In  the  large  circle  in  the  centre  a  small  brass  tube  is 
fixed,  which  serves  as  a  covering  for  the  upper  part  of  the 
middle  wire,  and  the  two  small  circles  are  to  receive  the 
wires  of  the  last  expansion. 

Fig,  213.  represents  the  bottom  of  the  exterior  tube,  in 
23 


266  THE    ELEMENTS    OF    MECHANICS.  CHAP.  XXI. 

which  the  small  circles  show  the  places  where  the  wires  of 
the  second  expansion  are  fastened,  and  the  larger  ones  the 
holes  for  the  other  pair  of  wires  to  pass  through. 

Fig.  214.  is  a  cylindrical  piece  of  brass,  showing  the  man- 
ner in  which  the  lower  ends  of  the  wires  of  the  last  expan- 
sion are  fastened  to  it,  and  the  hole  in  the  middle  is  that  by 
which  it  is  pinned  to  the  centre  of  the  bob.  The  upper  ends 
of  the  two  pair  of  wires  are,  as  we  have  observed,  fastened 
by  screwing  them  into  the  pieces  which  stop  up  the  ends  of 
the  tubes,  but  at  the  lower  ends  they  are  all  lixed  as  repre- 
sented in  jig.  214.  The  pieces  represented  by  Jigs.  213.  and 
214.  have  each  a  jointed  motion,  by  means  of  which  the  fel- 
low wires  of  each  pair  would  be  equally  stretched,  although 
they  were  not  exactly  of  the  same  length. 

The  action  of  this  pendulum  is  evidently  the  same  as  that 
of  the  gridiron  pendulum,  as  we  have  three  lengths  of  steel 
expanding  downwards,  and  two  of  brass  expanding  upwards. 
The  weight  of  the  pendulum  has  a  tendency  to  straighten 
the  steel  rods,  and  the  tubular  form  of  the  brass  compensa- 
tion effectually  precludes  the  fear  of  its  bending ;  an  advan- 
tage not  possessed  by  the  gridiron  pendulum,  in  which  brass 
rods  are  employed. 

Mr.  Troughton,  to  the  account  he  has  given  of  this  pen- 
dulum in  Nicholson's  Journal,  for  December,  1804,  has 
added  the  lengths  of  the  different  parts  of  which  it  was^com- 
posed,  and  the  expansions  of  brass  and  steel  from  which  these 
lengths  were  computed.  The  length  of  the  interior  tube 
was  31*9  inches,  and  that  of  the  exterior  one  32*8  inches,  to 
which  must  be  added  0-4,  the  quantity  by  which  in  this  pen- 
dulum the  centre  of  oscillation  is  higher  than  the  centre  of 
the  bob.  These  are  all  of  brass.  The  parts  which  are  of 
steel  are, — the  middle  wire,  which,  including  0'6,  the  length 
of  the  suspension  spring,  is  39-3  inches ;  the  first  pair  of 
wires  32-5  inches  ;  and  the  second  pair,  33-2  inches.  The 
expansions  used  were  for  brass  '00001666,  and  for  steel 
•00000661 ,  in  parts  of  their  length  for  one  degree  of  temper- 
ature. 

Benzcnberg's  Pendulum. 

This  pendulum  is  mentioned  in  Nicholson's  Journal,  for 
April,  1804,  and  is  taken  from  Voigt's  Magazin  fur  den  Neu- 
esten  Zustande  der  Naturkunde,  vol.  iv.  p.  787.  The  com- 
pensation appears  to  have  been  effected  by  a  single  rod  of 


CHAP.  xxi.  BENZENKERG'S  PENDULUM.  207 

lead  in  the  centre,  of  about  half  an  inch  thick ;  the  descend- 
ing rods  were  made  of  the  best  thick  iron  wire. 

As  this  pendulum  deserves  attention  from  the  ease  with 
which  it  may  be  made,  and  as  others  which  have  since  been 
produced  resemble  it  in  principle,  we  have  given  a  represen- 
tation of  it  vA.ji>g>  215.,  where  A  B  C  D  are  two  rods  of  iron 
wire  riveted  into  the  cross  pieces  A  C  B  D.  E  F  is  a  rod 
of  lead  pinned  to  the  middle  of  the  piece  B  D,  and  also  at 
its  upper  extremity  to  the  cross  piece  G  H,  into  which  the 
second  pair  of  iron  wires  are  fixed,  which  pass  downwards 
freely  through  holes  made  in  the  cross  piece  B  D.  The 
lower  extremities  of  these  last  iron  wires  are  fastened  into 
the  piece  K  L,  which  carries  the  bob  of  the  pendulum. 

To  determine  the  length  of  lead  necessary  for  the  compen- 
sation, we  must  recollect,  as  before,  that  the  distance  from 
the  point  of  suspension  to  the  centre  of  the  bob  (speaking 
always  of  a  pendulum  intended  to  vibrate  seconds)  must  be 
39  inches.  Lfet  us  suppose  the  total  length  of  the  iron  wire 
to  be  60  inches;  then,  from  the  table  which  we  have  given, 
we  have  -4308 -for  the  length  of  a  rod  of  lead,  the  expansion 
of  which  is  equivalent  to  that  of  an  iron  rod  whose  length  is 
unity.  Multiplying  60  inches  by  -4308,  we  have  25*84  inches 
of  lead,  which  would  compensate  60  inches  of  iron;  but 
this,  taken  from  60  inches,  leaves  only  34-16  instead  of  39 
inches.  Trying  again,  in  like  manner,  68*5  inches  of  iron, 
we  find  29-5  inches  of  lead  for  the  length,  affording  an  equiv- 
alent compensation,  and  which,  taken  from  68*5  inches, 
leaves  39  inches. 

The  length  of  the  rod  of  lead  then  required  as  a  compen- 
sation in  this"  pendulum  is  about  29£  inches. 

The  writer  of  this  article  would  suggest  another  form  for 
this  pendulum,  which  has  the  advantage  of  greater  simplici- 
ty of  construction. 

S  A,  fig.  216,  is  a  rod  of  iron  wire,  to  which  the  pendulum 
spring  is  attached.  Upon  this  passes  a  cylindrical  tube  of 
lead,  294-  inches  long,  which  is  either  pinned  at  its  lower  ex- 
treniity  to  the  end  of  the  iron  Tod  S  A,  or  rests  upon  a  nut 
firmly  screwed  upon  the  extremity  of  this  rod. 

A  tube  of  sheet  iron  passes  over  the  tube  of  lead,  and  is 
furnished  at  top  with  a  ftanche,  by  which  it  is  supported  upon 
the  leaden  tube ;  or  it  may  be  fastened  to  the  top  of  this  tube 
in  any  manner  that  may'  be  thought  convenient. 

Tne  bob  of  the  pendulum  may  be  either  passed  upon  the 


268  THE  ELEMENTS  OF  MECHANICS.      CHAP.  XXI. 

iron  tube  (continued  to  a  sufficient  length)  and  secured  by  a 
pin  passing  through  the  centre  of  the  bob,  or  the  iron  tube 
may  be  terminated  by  an  iron  wire  serving  the  same  purpose. 
Here  we  have  evidently  the  same  expansions  upwards  and 
downwards  as  in  the  gridiron  form,  given  to  this  pendulum 
by  Mr.  Benzenberg,  joined  to  the  compactness  of  Troughton's 
tubular  pendulum. 

Ward's  Compensation  Pendulum. 

In  the  year  1806,  Mr.  Henry  Ward,  of  Blandford  in  Dor- 
setshire, received  the  silver  medal  of  the  Society  of  Arts  for 
the  compensation  pendulum  which  we  are  about  to  describe. 

Fig.  217.  is  a  side  view  of  the  pendulum  rod  when  togeth- 
er. H  H  and  I  I  are  two  flat  rods  of  iron  about  an  eighth 
of  an  inch  thick.  K  K  is  a  bar  of  zinc  placed  between 
them,  and  is  nearly  a  quarter  of  an  inch  thick.  The  corners 
of  the  iron  bars  are  bevelled  off,  which  gives  them  a  much 
lighter  appearance.  These  bars  are  kept  together  by  means 
of  three  screws,  O  O  O,  which  pass  through  oblong  holes  in 
the  bars  H  H  and  K  K,  and  screw  into  the  rod  I  I.  The 
bar  H  H  is  fastened  to  the  bar  of  zinc  K  K,  by  the  screw  m, 
which  is  called  the  adjusting  screw.  This  screw  is  tapped 
into  H  H,  and  passes  just  through  K  K  ;  but  that  part  of  the 
screw  which  passes  K  K  has  its  threads  turned  off.  The 
iron  bar  I  I  has  a  shoulder  at  its  upper  end,  and  rests  on  the 
top  of  the  zinc  bar  K  K,  and  is  wholly  supported  by  it. 
There  are  several  holes  for  the  screw  w,  in  order  to  adjust 
the  compensation. 

The  action  of  this  pendulum  is  similar  to  that  last  de- 
scribed, the  zinc  expanding  upwards  as  much  as  the  iron 
rods  expand  downwards,  and  consequently  the  distance 
from  the  point  of  suspension  to  the  centre  of  oscillation 
remains  the  same. 

Mr.  Ward  states  that  the  expansion  of  the  zinc  he  used 
(hammered  zinc)  was  greater  than  that  given  in  the  tables. 
He  found  that  the  true  length  of  the  zinc  bar  should  be 
about  23  inches :  our  computation  would  make  it  nearly  26. 

The  Compensation  Tube  of  Julien  le  Roy. 

We  mention  this  merely  to  state  that  it  is  similar  in  prin- 
ciple to  the  apparatus  represented  at  fig.  204.,  with  merely 


CHAP.  xxi.  RATER'S  PENDULUM.  209 

this  difference,  that,  instead  of  the  steel  rod  being  fixed  to 
a  cross  piece  proceeding  from  the  brass  bar  B  R,  it  is  at- 
tached to  a  cap  fitted  upon  a  brass  tube  (through  which  it 
passes)  of  the  same  length  as  that  of  the  brass  rod  B  R. 
Cassini  spoke  well  of  this  pendulum,  and  it  was  used  in  the 
observatory  of  Cluny  about  the  year  1748. 

Deparcieutfs  Compensation. 

This  was  contrived  in  the  same  year  as  that  invented  by 
Jnlien  le  Roy.  It  is  represented  atj%-.  218,  where  A  B  D  F 
is  a  steel  bar,  the  ends  of  which  are  to  be  fixed  to  the  lower 
sides  of  pieces  forming  a  part  of  the  cock  of  the  pendulum. 
G  E  I  H  is  of  brass,  and  stands  with  its  extremities  resting 
on  the  horizontal  part  B  D  of  the  steel  frame.  The  upper 
part  E  I  of  the  brass  frame  passes  above  the  cock  of  the  pen- 
dulum, and  admits  the  tapped  wire  K,  to  which  the  pendulum 
spring  is  fixed  through  a  squared  hole  in  the  middle.  A  nut 
upon  this  tapped  wire  gives  the  adjustment  for  time.  The 
spring  passes  through  the  slit  in  the  cock  in  the  usual 
manner. 

It  may  be  easily  perceived  that  this  pendulum  is 'in1  princi- 
ple the  same  as  that  of  Le  Roy  ;  the  expansion  of  the  total 
length  of  steel  A  B  S  C  downwards  being  compensated  by 
the  equivalent  expansion  of  the  brass  bar  G  E  upwards.  It 
is,  however,  preferable  to  Le  Roy's,  because  the  compensa- 
tion is  contained  in  the  clock  case. 

Deparcieux  had  previously  published,  in  the  year  1739,  an 
improvement  of  an  imperfectly  compensating  '''pendulum, 
proposed  in  the  year  1733  by  Regnauld,  a  clockmaker  of 
Chalons.  In  this  pendulum  Deparcieux  employed  a  lever 
with  unequal  arms  to  increase  the  effect  of  the  expansion  of 
the  brass  rod ,  which  was  too  short. 

We  may  here  remark,  that  all  fixed  compensations  are 
liable  to  the  same  objection,  namely,  that  of  not  moving  with 
the  pendulum,  and  therefore  not  taking  precisely  the  same 
temperature. 

Captain 'Kdter's  Compensation  iFVndulum. 

In  Nicholson's  Journal,  for  July,  1808,  is  the  description 
of  a  compensation  pendulum  by  the  writer  of  this  article. 
In  this  pendulum  the  rod  is  of  white  deal,  three  quarters  of 
23  *  ha  •!,-. •'.•::-.;: 


270  THE  ELEMENTS  OF  MECHANICS.    CHAP.  XXI. 

an  inch  wide,  and  a  quarter  of  an  inch  thick.  It  was  placed 
in  an  oven,  and  suffered  to  remain  there  for  a  long  time  until 
it  became  a  little  charred.  The  ends  were  then  soaked  in 
melted  sealing-wax;  and  the  rod,  being  cleaned,  was  coated 
several  times  with  copal  varnish.  To  the  lower  extremity  of 
the  rod  a  cap  of  brass  was  firmly  fixed,  from  which  a  strong 
steel  screw  proceeded  for  the  purpose  of  regulating  the  pen- 
dulum for  time  in  the  usual  manner. 

A  square  tube  of  zinc  was  cast,  seven  inches  long,  and 
three  quarters  of  an  inch  square ;  the  internal  dimensions 
being  four  tenths  of  an  inch.  The  lower  part  of  the  pendu- 
lum rod  was  cut  away  on  the  two  sides,  so  as  to  slide  with 
perfect  freedom  within  the  tube  of  zinc.  To  the  bottom 
of  this  zinc  tube  a  piece  of  brass  a  quarter  of  an  inch  thick 
was  soldered,  in  which  a  circular  hole  was  made  nearly  four 
tenths  of  an  inch  in  diameter,  having  a  screw  on  the  inside. 
A  cylinder  of  zinc,  furnished  with  a  corresponding  screw  on 
its  surface,  fitted  into  this  aperture,  and  a  thin  plate  of  brass 
screwed  upon  the  cylinder,  served  as  a  clamp  to  prevent  any 
shake  after  the  length  of  zinc  necessary  for  compensation 
should  have  been  determined.  A  hole  was  made  through  the 
axis  of  the  cylinder,  through  which  passed  the  steel  screw 
terminating  the  pendulum  rod. 

An  opening  was  made  through  the  bob  of  the  pendulum, 
extending  to  its  centre,  to  admit  the  square  tube  of  zinc 
which  was  fixed  at  its  upper  extremity  to  the  centre  of  the 
bob.  The  pendulum  rod  passed  through  the  bob  in  the  usual 
manner,  and  the  whole  was  supported  by  a  nut  on  the  steel 
screw  at  the  extremity. 

In  this  form,  the  compensation  acts  immediately  upon  the 
centre  of  the  bob,  elevating  it  along  the  rod  as  much  as  the 
rod  elongates  downwards :  the  method  of  calculating  the 
length  of  the  required  compensation  is  precisely  the  same  as 
that  we  have  before  given. 

Assuming  the  length  of  the  deal  rod  to  be  43  inches,  and 
multiplying  this  by  -1313  from  Table  II.,  we  have  5-64  inches 
for  the  length  of  the  zinc  necessary  to  counteract  the  expan- 
sion of  the  deal.  The  length  of  the  steel  screw,  between  the 
termination  of  the  pendulum  rod  and  the  nut,  was  two  inches, 
and  that  of  the  suspension  spring  one  inch.  Now,  3  inches  of 
steel  multiplied  by  -3682  wouJd  give  1-10  inch  for  the  length 
of  zinc  which  would  compensate  the  steel,  and,  adding  this 
to  5-64  inches,  we  have  6-74  inches  fi>r  the  whole  length  of 
zinc  required. 


CHAP.  xxi.  REID'S  PENDULUM.  271 

In  this  pendulum,  the  length  of  the  compensating  part  may 
be  varied  by  means  of  the  zinc  cylinder  furnished  with  a 
screw  for  that  purpose.  The  bob  of  this  pendulum  and  its 
compensation  are  represented  at  Jig.  219. 

It  has  been  objected  to  the  use  of  wooden  pendulum  rods, 
that  it  is  difficult,  if  not  impossible,  to  secure  them  from  the 
action  of  moisture,  which  would  at  once  be  fatal  to  their  cor- 
rect performance.  The  pendulum  now  before  us  has,  how- 
ever, been  going  with  but  little  intermission  since  it  was  first 
constructed  :  it  is  attached  to  a  sidereal  clock,  not  of  a  supe- 
rior description,  and  exposed  to  very  considerable  variations 
of  moisture  and  dryness ;  yet  the  change  in  its  rate  has  been 
so  very  trifling  as  to  authorize  the  belief,  that  moisture  has  lit- 
tle or  no  effect  upon  a  wooden  rod  prepared  in  the  manner 
we  have  described.  Its  rate,  under  different  temperatures, 
tliows  that  it  is  over-compensated;  the  length  of  the  zinc 
remaining,  as  stated  in  Nicholson's  Journal,  7-42  inches, 
instead  of  which  it  appears,  by  our  present  compensation, 
that  it  should  be  6-78  inches. 

Reid's  Condensation  Pendulum, 

Mr.  Adam  Heid  of  Woolwich  presented  to  the  Society  of 
Arts,  in  1809,  a  compensation  pendulum,  for  which  he  was 
rewarded  with  fifteen  guineas.  This  pendulum  is  the  same 
in  principle  with  that  last  described ;  the  rod,  however,  is  of 
steel  instead  of  wood,  and  the  compensation  possesses  no 
means  of  adjustment.  This  pendulum  is  represented  at  Jig. 
220.,  where  S  B  is  the  steel  rod,  a  little  thicker  where  it  en- 
ters the  bob  C,  and  of  a  lozenge  shape  to  prevent  the  bob 
turning,  but  above  and  below  it  is  cylindrical. 

A  tube  of  zinc  D  passes  to  the  centre  of  the  bob  from 
below,  and  the  bob  is  supported  upon  it  by  a  piece  which 
crosses  its  centre,  and  which  meets  the  upper  end  of  the  tube. 

The  rod  being  passed  through  the  bob  and  zinc  tube,  a  nut 
is  applied  upon  a  screw  at  the  lower  extremity  of  the  rod  in 
the  usual  manner.  If  the  compensation  should  be  too  much, 
the  zinc  tube  is  to  be  shortened  until  it  is  correct. 

The  length  of  the  zinc  tube  will  be  the  same  in  this  pen- 
dulum as  in  that  of  Mr.  Ward — about  23  inches,  if  his  ex- 
periments are  to  be  relied  upon. 

The  objection  to  this  pendulum  appears  to  be  its  great 
lengtri,  which  amounts  to  62  inches.  We  conceive  it  would 


272  THE  ELEMENTS  OF  MECHANICS.     CHAP.  XXI 

be  preferable  to  place  the  zinc  above  the  bob,  as  in  the  modi- 
fication which  we  have  suggested  of  Benzenberg's  pendulum. 

Elticott's  Pendulum. 

It  appears  that  the  idea  of  combining  the  expansions  of 
different  metals  with  a  lever,  so  as  to  form  a  compensation 
pendulum,  originated  with  Mr.  Graham  :  for  Mr.  Short,  in 
the  Philosophical  Transactions  for  1752,  states  that  he  was 
informed  by  Mr.  Shelton,  that  Mr.  Graham,  in  the  year  1737, 
made  a  pendulum,  consisting  of  three  bars,  one  of  steel  be- 
tween two  of  brass  ;  and  that  the  steel  bar  acted  upon  a  lever 
so  as  to  raise  the  pendulum  when  lengthened  by  heat,  and  to 
let  it  down  when  shortened  by  cold. 

This  pendulum,  however,  was  found  upon  trial  to  move  by 
jerks,  and  was  therefore  laid  aside  by  the  inventor  to  mak 
way  for  the  mercurial  pendulum. 

Mr.  Short  also  says,  that  Mr.  Fotheringham,  a  Quaker  c/ 
Lincolnshire,  caused  a  pendulum  to  be  made,  in  the  year 
1738  or  1739,  consisting  of  two  bars,  one  of  brass  and  the 
other  of  steel,  fastened  together  by  screws,  with  levers  to  raise 
or  let  down  the  bob,  and  that  these  levers  were  placed  above 
the  bob. 

Mr.  John  Ellicott  of  London  had  made  very  accurate  ex- 
periments on  the  relative  expansions  of  seven  different  metals, 
which,  however,  will  be  found  to  differ  more  or  less  from  the 
results  of  the  experiments  of  others.  It  is  not,  however,  from 
this  to  be  concluded  that  EHicott's  determinations  were  erro- 
neous ;  for  the  expansion  of  'a  metal  will  suffer  considerable 
change  even  by  the  processes  to  which  it!  is  necessarily  sub- 
jected jn  the  construction  of  a  pendulum.  It  is  therefore 
desirable,  whenever  a' compensation  pendulum  is  to 'be- made, 
that  the  expansions  of  the  materials,  employed  should  be 
determined  after  the  processes  of  drilling;  filing  and  ham- 
mering have  been  gone  through. 

It  has  bejen  objected  to  Harrison's  gridiron  pendulum,  that 
the  adjustment  of  the  rods  was  inconvenient,  and  that  the 
expansion  of  the  bob  supported  at  its  lower  edge  would  „  un- 
less taken  into  the  account,  vitiate  the  compensation.  These 
considerations,  it  is  supposed,  gave  risfe  to  Elh'cott's  pendu- 
lum, which  is  nearly  similar  to  those  we  'have  just  mentioned. 

EHicott's  pendulum  is  thus.'  constructed  :- — A  bar  of  brass 
and  a  bar  of  iron  are  firmly  fixed  together  at  their  upper  ends, 


CHAP.    XXi.  COMPENSATION    PENDULUM.  273 

the  bar  of  brass  lying  upon  the  bar  of  iron,  which  is  the  rod 
of  the  pendulum.  These  bars  are  held  near  each  other  by 
screws  passing  through  oblong  holes  in  the  brass,  and  tapped 
into  the  iron,  and  thus  the  brass  is  allowed  to  expand  or  con- 
tract freely  upon  the  iron  with  any  change  of  temperature. 
The  brass  bar  passes  to  the  centre  of  the  bob  of  the  pendu- 
lum, a  little  above  and  below  which  the  iron  is  left  broader 
for  the  purpose  of  attaching  the  levers  to  it,  and  the  iron  is 
made  of  a  sufficient  length  to  pass  quite  through  the  bob 
of  the  pendulum. 

The  pivots  of  two  strong  steel  levers  turn  in  two  holes 
drilled  in  the  broad  part  of  the  iron  bar.  The  short  arms  of 
these  levers  are  in  contact  with  the  lower  extremity  of  the 
brass  bar,  and  their  longer  arms  support  the  bob  of  the  pendu- 
lum by  meeting  the  heads  of  two  screws  which  pass  horizon- 
tally from  each  side  of  the  bob  towards  its  centre.  By  ad- 
vancing these  screws  towards  the  centre  of  the  bob,  the 
longer  arms  of  the  lever  are  shortened,  and  thus  the  compen- 
sation may  be  readily  adjusted.  At  the  lower  end  of  the  iron 
rod,  under  the  bob,  a  strong  double  spring  is  fixed,  to  support 
the  greater  part  of  the  weight  of  the  bob  by  its  pressure  up- 
wards against  two  points  at  equal  distances  from  the  pendu- 
lum rod.  Mr.  Ellicott  gave  a  description  of  this  pendulum 
to  the  Royal  Society  in  1752,  but  he  says  the  thought  was 
executed  in  1738.  As  this  pendulum  is  very  seldom  met 
with,  we  think  it  unnecessary  to  give  a  representation  of  it. 

Compensation  by  Means  of  a  Compound  Bar  of  Steel  and 
Brass. 

Several  compensations  for  pendulums  have  been  proposed, 
by  means  of  a  compound  bar  formed  of  steel  and  brass  sol- 
dered together.  In  a  bar  of  this  description,  the  brass  expand- 
ing more  than  the  steel,  the  bar  becomes  curved  by  a  change 
of  temperature,  the  brass  side  becoming  convex  and  the  steel 
concave  with  heat.  Now,  if  a  bar  of  this  description  have  its 
ends  resting  on  supports  on  each  side  the  cock  of  the  pen- 
dulum, the  bar  passing  above  the  cock  with  the  brass  upper- 
most, if  the  pendulum  spring  be  attached  to  the  middle  of  the 
bar,  and  it  pass  in  the  usual  manner  through  the  slit  of  the 
cock,  it  is  evident  that,  by  an  increase  of  temperature,  the 
bar  will  become  curved  upwards,  and  the  pendulum  spring 
be  drawn  upwards  through  the  slit,  and  thus  the  elongation 


274  THE  ELEMENTS  OF  MECHANICS.  CHAP.  XXI 

of  the  pendulum  downwards  wiU  be  compensated.  The 
compensation  may  be  adjusted  by  varying  the  distance  of  the 
points  of  support  from  the  middle  of  the  bar. 

Such  was  one  of  the  modes  of  compensation  proposed  by 
Nicholson.  Others  of  the  same  description  (that  is,  with 
compound  bars)  have  been  brought  before  the  public  by  Mr. 
Thomas  Doughty  and  Mr.  David  Ritchie;  but  as  they  are 
supposed  to  be  liable  to  many  practical  objections,  we  do  not 
think  it  requisite  to  describe  them  more  particularly. 

There  is,  however,  a  mode  of  compensation  by  means  of  a 
compound  bar,  described  by  M.  Biot  in  the  first  volume  of  his 
Traite  de  Physique,  which  appears  to  possess  considerable 
merit,  of  which  he  mentions  having  first  witnessed  the  suc- 
cessful employment  by  the  inventor,  a  clockmaker  named 
Martin.  At  Jig,  2£l.,  S  C  is  the  rod  of  the  pendulum,  made 
in  the  usual  manner,  of  iron  or  steel ;  this  rod  passes  through 
the  middle  of  a  compound  bar  of  brass  and  steel  (the  brass 
being  undermost),  which  should  be  furnished  with  a  short  tube 
and  screws,  by  means  of  which,  or  by  passing  a  pin  through 
the  tube  and  rod,  it  may  be  securely  fixed  at  any  part  of  the 
pendulum  rod. 

Two  small  equal  weights  W  W  slide  along  the  compound 
bar,  and,  when  their  proper  position  has  been  determined, 
may  be  securely  clamped.  ',""J^. 

The  manner  in  which  this  compensation  acts  is  thus  : — Sup- 
pose the  temperature  to  increase,  the  brass  expanding  more 
than  the  steel,  the  bar  becomes  curved,  and  its  extremities 
carrying  the  weights  W  and  W  are  elevated,  and  thus  the 
place  of  the  centre  of  oscillation  is  made  to  approach  the 
point  of  suspension  as  much,  when  the  compensation  is  prop- 
erly adjusted,  as  it  had  receded  from  it  by  the  elongation 
of  the  pendulum  rod. 

There  are  three  methods  of  adjusting  this  compensation  : 
the  first,  by  increasing  or  diminishing  the  weights  W  and  W  ; 
the  second,  by  varying  the  distance  of  the  weights  W  and 
W  from  the  middle  of  the  bar;  and  the  third,  by  varying  the 
distance  of  the  bar  from  the  bob  of  the  pendulum,  taking 
care  not  to  pass  the  middle  of  the  rod.  The  effect  of  the 
compensation  is  greater  as  the  weights  W  and  W  are  greater 
or  more  distant  from  the  centre  of  the  bar,  and  also  as  the 
bar  is  nearer  to  the  bob, of  the  pendulum. 

M.  Biot  says  that  he  and  M.  Matthieu  employed  a  pendu- 
um  of  this  kind  for  a  long  time  in  making  astronomical  ob 


CHAP.  XXI.  MERCURIAL  PENDULUM.  275 

servations,  in  which  they  were  desirous  of  attaining  an  ex- 
treme degree  of  precision,  and  that  they  found  its  rate  to  be 
always  perfectly  regular. 

In  all  the  pendulums  which  we  have  described,  the  bob  is 
supposed  to  be  fixed  to  the  rod  by  a  pin  passing  through  its 
centre,  and  the  adjustment  for  time  is  to  be  made  by  means 
of  a  small  weight  sliding  upon  the  rod. 

Of  the  Mercurial  Pendulum. 

We  have  been  guided,  in  our  arrangement  of  the  pendu- 
lums which  we  have  described,  by  the  similarity  in  the  mode 
of  compensation  employed ;  and  we  have  now  to  treat  of 
that  method  of  compensation  which  is  effected  by  the  expan- 
sion of  the  material  of  which  the  bob  itself  of  the  pendulum 
is  composed. 

On  this  subject,  as  we  have  before  observed,  an  admirable 
paper,  from  the  pen  of  Mr.  Francis  Baily,  may  be  found  in 
the  Memoirs  of  the  Astronomical  Society  of  London,  which 
leaves  nothing  to  be  desired  by  the  mathematical  reader.  But 
as  our  object  is  to  simplify,  and  to  render  our  subjects  as 
popular  as  may  be,  we  must  endeavor  to  substitute  for  the 
perfect  accuracy  which  Mr.  Baily's  paper  presents,  such  rules 
as  may  be  found  not  only  readily  intelligible,  but  practically 
applicable,  within  the  limits  of  those  inevitable  errors  which 
arise  from  a  want  of  knowledge  of  the  exact  expansion  of  the 
materials  employed. 

At  Jig.  22*2.,  let  S  B  represent  the  rod  of  a  pendulum,  and 
F  C  B  a  metallic  tube  or  cylinder,  supported  by  a  nut  at  the 
extremity  of  the  pendulum  rod,  in  the  usual  manner,  and 
having  a  greater  expansibility  than  that  of  the  rod.  Now 
C,  the  centre  of  gravity,  supposing  the  rod  to  be  without 
weight,  will  be  in  the  middle  of  the  cylinder  ;  and  if  C  B,  or 
half  the  cylinder,  be  of  such  a  length  as  to  expand  upwards 
as  much  as  the  pendulum  rod  S  B  expands  downwards,  it  is 
evident  that  the  centre  of  gravity  C  will  remain,  under  any 
change  of  temperature,  at  the  same  distance  from  the  point 
of  suspension  S.  M.  Biot  imagined  that,  in  effecting  this,  a 
compensation  sufficiently  accurate  would  be  obtained  ;  but 
Mr.  Baily  has  shown  that  this  is  by  no  means  the  fact. 

Let  us  suppose  the  place  of  the  centre  of  oscillation  to 
be  at  O,  about  three  or  four  tenths  of  an  inch,  in  a  pendu- 
lum of  the  usual  construction,  below  the  centre  of  gravity. 


276  THE    ELEMENTS    OP   MECHANICS.  CHAP.  XXI 

Now,  the  object  of  the  compensation  is  to  preserve  the  dis- 
tance from  S  to  O  invariable,  and  not  the  distance  from 
S  toC. 

The  distance  of  the  centre  of  oscillation  varies  with  the 
length  of  the  cylinder  F  B,  and  hence  suffers  an  alteration 
in  its  distance  from  the  point  of  suspension  by  the  elongation 
of  the  cylinder,  although  the  distance  of  the  centre  of  gravity 
C  from  the  point  of  suspension  remains  unaltered. 

We  shall  endeavor  to  render  this  perfectly  familiar.  Sup- 
pose a  metallic  cylinder,  6  inches  long,  to  be  suspended  by  a 
thread  36  inches  long,  thus  forming  a  pendulum  in  which 
the  distance  of  the  centre  of  gravity  from  the  point  of  sus- 
pension is  39  inches  :  the  centre  of  oscillation  in  such  a  pen- 
dulum will  be  nearly  one  tenth  of  an  inch  below  the  centre 
of  gravity.  Now,  let  us  imagine  cylindrical  portions  of  equal 
lengths  to  be  added  to  each  end  of  the  cylinder,  until  it 
reaches  the  point  of  suspension  ;  we  shall  then  have  a  cylin- 
der of  78  inches  in  length,  the  centre  of  gravity  of  which 
will  still  be  at  the  distance  of  39  inches  from  the  point  of 
suspension.  But  it  is  well  known  that  the  centre  of  oscilla- 
tion of  such  a  cylinder  is  at  the  distance  of  about  two  thirds 
of  its  length  from  the  point  of  suspension.  The  centre  of 
oscillation,  therefore,  has  been  removed,  by  the  elongation  of 
the  cylinder,  about  13  inches  below  the  centre  of  gravity, 
whilst  the  centre  of  gravity  has  remained  stationary, 

Now,  the  same  thing  as  that  which  we  have  just  described 
takes  place,  though  in  a  very  minor  degree,  with  our  for- 
mer cylinder,  employed  as  a  compensating  bob  to  a  pendulum. 
The  rod  expands  downwards,  the  centre  of  gravity  remains 
at  the  same  distance  from  the  point  of  suspension,  and  the 
cylinder  elongates  both  above  and  below  this  point ;  the  con- 
sequence of  which  is,  that  though  the  centre  of  gravity  has 
remained  stationary,  the  distance  of  the  centre  of  oscillation 
from  the  point  of  suspension  has  increased.  It  is,  therefore, 
evident  that  the  length  of  the  compensation  must  be  such  as 
to  carry  the  centre  of  gravity  a  little  nearer  to  the  point  of 
suspension  than  it  was  before  the  expansion  took  place ;  by 
which  means  the  centre  of  oscillation  will  be  restored  to  its 
former  distance  from  the  point  of  suspension. 

Let  us  suppose  the  expansions  to  have  taken  place,  and  that, 
the  centre  of  gravity  remaining  at  the  same  distance  from 
the  point  of  suspension,  the  centre  of  oscillation  is  removed 
to  a  greater  distance,  as  we  have  before  explained.  It  is 


CHAP.  xxi.       GRAHAM'S  PENDULUM.  277 

well  known  that  the  product  obtained  by  multiplying  the  dis- 
tance from  the  point  of  suspension  to  the  centre  of  gravity, 
by  the  distance  from  the  centre  of  gravity  to  the  centre  of 
oscillation,  is  a  constant  quantity ;  if,  therefore,  the  distance 
from  the  centre  of  gravity  to  the  point  of  suspension  be 
lessened,  the  distance  from  the  centre  of  gravity  to  the  cen- 
tre of  oscillation  will  be  proportionally,  though  not  equally, 
increased,  and  the  centre  of  oscillation  will,  therefore,  be 
elevated.  We  see,  then,  if  we  elevate  the  centre  of  gravity 
precisely  the  requisite  quantity,  by  employing  a  sufficient 
length  of  the  compensating  material,  that  although  the  dis- 
tance from  the  centre  of  gravity  to  the  point  of  suspension 
is  lessened.,  yet  the  distance  from  the  point  of  suspension 
to  the  centre  of  oscillation  will  suffer  no  change. 

The  following  rule  for  finding  the  length  of  the  compen- 
sating material,  in  a  pendulum  of  the  kind  we  have  been  con- 
sidering, will  be  found  sufficiently  accurate  for  all  practical 
purposes : — 

Find,  in  the  manner  before  directed,  the  length  of  the  com- 
pensating material,  the  expansion  of  which  will  be  equal  to 
that  of  the  rod  of  the  pendulum.  Double  this  length,  and 
increase  the  product  by  its  one  tenth  part,  which  will  give  the 
total  length  required.  We  shall  give  examples  of  this  as  we 
proceed. 

Graham's  Mercurial  Pendulum. 

It  was  in  the  year  1721  that  Graham  first  put  up  a  pendu 
lum  of  this  description,  and  subjected  it  to  the  test  of  ex- 
periment ;  but  it  appears  to  have  been  afterwards  set  aside 
to  make  way  for  Harrison's  gridiron  pendulum,  or  for  others 
of  a  similar  description.  For  some  years  past,  however,  its 
merits  have  been  more  generally  known,  and  it  is  not  sur- 
prising that  it  should  be  considered  as  preferable  to  others, 
both  from  the  simplicity  of  its  construction,  and  the  perfect 
ease  with  which  the  compensation  may  be  adjusted. 

We  have  already  alluded  to  Mr.  Baily's  very  able  paper  on 
this  pendulum,  and  we  shall  take  the  liberty  of  extracting 
from  it  the  following  description  : — 

At  Jig.  223.  is  a  drawing  of  the  mercurial  pendulum,  as 
constructed  in  the  manner  proposed  by  Mr.  Baily. 

"  The  rod  S  F  is  made  of  steel,  and  perfectly  straight ;  its 
24 


278  THE  ELEMENTS  OF  MECHANICS.     CHAP.  XXI. 

form  may  be  either  cylindrical,  of  about  a  quarter  of  an  inch 
in  diameter,  or  a  flat  bar,  three  eighths  of  an  inch  wide, 
and  one  eighth  of  an  inch  thick  :  its  length  from  S  to  F, 
that  is,  from  the  bottom  of  the  spring  to  the  bottom  of 
the  rod  at  F,  should  be  34  inches.  The  lower  part  of 
this  rod,  which  passes  through  the  top  of  the  stirrup, 
and  about  half  an  inch  above  and  below  the  same,  must 
be  formed  into  a  coarse  and  deep  screw,  about  two  tenths 
of  an  inch  in  diameter,  and  having  about  thirty  turns  in 
an  inch.  A  steel  nut  with  a  milled  head  must  be  placed 
at  the  end  of  the  rod,  in  order  so  support  the  stirrup ;  and  a 
similar  nut  must  also  be  placed  on  the  rod  above  the  head  of 
the  stirrup,  in  order  to  screw  firmly  down  on  the  same,  and 
thus  secure  it  in  its  position,  after  it  has  been  adjusted  nearly 
to  the  required  rate.  These  nuts  are  represented  at  B  and  C 
A  small  slit  is  cut  in  the  rod,  where  it  passes  through  the 
head  of  the  stirrup,  through  which  a  steel  pin  E  is  screwed, 
in  order  to  keep  the  stirrup  from  turning  round  on  the  rod. 
The  stirrup  itself  is  also  made  of  steel,  and  the  side-pieces 
should  be  of  the  same  form  as  the  rod,  in  order  that  they 
may  readily  acquire  the  same  temperature.  The  top  of  the 
stirrup  consists  of  a  flat  piece  of  steel,  shaped  as  in  the  draw- 
ing, somewhat  more  than  three  eighths  of  an  inch  thick. 
Through  the  middle  of  the  top  (which  at  this  part  is  about 
one  inch  deep)  a  hole  must  be  drilled  sufficiently  large  to 
enable  the  screw  of  the  rod  to  pass  freely,  but  without  shak- 
ing. The  inside  height  of  the  stirrup  from  A  to  D  may  be 
8£  inches,  and  the  inside  width  between  the  bars  about  three 
inches.  The  bottom  piece  should  be  about  three  eighths 
of  an  inch  thick,  and  hollowed  out  nearly  a  quarter  of  an 
inch  deep,  so  as  to  admit  the  glass  cylinder  freely.  This 
glass  cylinder  should  have  a  brass  or  iron  cover  G,  which 
should  fit  the  mouth  of  it  freely,  with  a  shoulder  projecting 
on  each  side,  by  means  of  which  it  should  be  screwed  to  the 
side-bars  of  the  stirrup,  and  thus  be  secured  always  in  the 
same  position.  This  cap  should  not  press  on  the  glass  cylin- 
der, so  as  to  prevent  its  expansion.  The  measures  above 
given  may  require  a  slight  modification,  according  to  the 
weight  of  the  mercury  employed,  and  the  magnitude  of  the 
cylinder :  the  final  adjustment,  however,  may  be  safely  left 
to  the  artist.  Some  persons  have  recommended  that  a  circu- 
lar piece  of  thick  plate  glass  should  float  on  the  mercury,  in 


CHAP.  XXI.    BAILY  ON  GRAHAM'S  PENDULUM.  279 

order  to  preserve  its  surface  uniformly  level.*  The  part  at 
the  bottom  marked  H  is  a  piece  of  brass  fastened  with 
screws  to  the  front  of  the  bottom  of  the  stirrup,  through  a 
small  hole,  in  which  a  steel  wire  or  common  needle  is  passed, 
in  order  to  indicate  (on  a  scale  affixed  to  the  case  of  the 
clock)  the  arc  of  vibration.  This  wire  should  merely  rest 
in  the  hole,  whereby  it  may  be  easily  removed  when  it  is  re- 
quired to  detach  the  pendulum  from  the  clock,  in  order  that 
the  stirrup  might  then  stand  securely  on  its  base.  One  of 
the  screw  holes  should  be  rather  larger  than  the  body  of 
the  screw,  in  order  to  admit  of  a  small  adjustment,  in  case 
the  steel  wire  should  not  stand  exactly  perpendicular  to  the 
axis  of  motion.  The  scale  should  be  divided  into  degrees, 
and  not  inches,  observing  that  with  a  radius  of  44  inches  (the 
estimated  distance  from  the  bend  of  the  spring  to  the  end  of 
the  steel  wire)  the  length  of  each  degree  on  the  scale  must 
be  0-768  inch." 

In  order  to  determine  the  length  of  the  mercurial  column 
necessary  to  form  the  compensation  for  this  pendulum,  we 
must  proceed  in  the  following  manner  : — 

Let  us  suppose  the  length  of  the  steel  rod  and  stirrup  to- 
gether to  be  42  inches.  The  absolute  expansion  of  the 
mercury  is  -00010010 ;  but  it  is  not  the  absolute  expansion, 
but  the  vertical  expansion  in  a  glass  cylinder,  which  is  re- 
quired, and  this  will  evidently  be  influenced  by  the  expansion 
of  the  base  of  the  cylinder.  It  is  easily  demonstrable  that, 
if  we  multiply  the  linear  expansion  of  any  substance  (always 
supposed  to  be  a  very  small  part  of  its  length)  by  3>  we  may 
in  all  cases  take  the  result  for  the  cubical  or  absolute  expan- 
sion of  such  substance.  In  like  manner,  if  we  multiply  the 
linear  expansion  by  2,  we  shall  have  the  superficial  expan- 
sion. 

If  we  want  the  apparent  expansion  of  mercury,  the  abso- 
lute or  cubical  expansion  of  the  glass  vessel  must  be  deduct- 
ed from  the  absolute  expansion  of  the  mercury,  which  will 
leave  its  excess  or  apparent  expansion.  In  like  manner, 

*  The  variation  produced  in  Ihe  height  of  the  column  of  mercury  (supposed 
to  be  6-5  inches  high)  by  an  alteration  of  ±  16°  in  the  temperature  will  be 
only  dt  O'Ol  of  an  inch,  or,  in  other  words,  0-01  of  an  inch  will  be  the  total 
variation  from  its  mean  state,  by  an  alteration  of  32°  in  the  temperature.  It  is 
therefore  probable  that  inmost  cases  of  moderate  alteration  in  the  temperature, 
the  centre  only  of  the  column  of  mercury  is  subject  to  elevation  and  depres- 
sion, whilst  the  exterior  parts  remain  attached  to  the  sides  of  the  glass  vessel. 
It  was  with  a  view  to  obviate  this  inconvenience  that  Henry  Browne,  Esq. 
of -Portland  Place  (I  believe)  first  suggested  the  piece  of  floating  glass. 


280  THE    ELEMENTS    OF    MECHANICS.  CHAP.  XXI. 

deducting  the  superficial  expansion  of  glass  from  the  abso- 
lute expansion  of  mercury,  we  shall  have  its  relative  vertical 
expansion.  Now,  taking  the  rate  of  expansion  of  glass  to 
be  -00000479,  and  multiplying  it  by  2,  the  relative  vertical 
expansion  of  the  mercury  in  the  glass  cylinder  will  be 
•00010010— -00000958  =  -00009052. 

The  expansion  of  a  steel  rod,  according  to  our  table,  is 
•0000063596  ;  which,  divided  by  -00009052,  gives  -0703  for 
the  length  of  a  column  of  mercury,  the  expansion  of  which  is 
equal  to  that  of  a  steel  rod  whose  length  is  unity. 

We  have  now  to  multiply  42  inches  by  -0703,  which  gives 
2-95  inches;  and  this,  deducted  from  42,  leaves  39*1  inches; 
so  that  the  length  of  rod  we  have  chosen  is  sufficiently  near 
the  truth.  Now,  double  2-95  inches,  and  add  one  tenth  of 
its  product,  and  we  shall  have  6-49  inches  for  the  length  of 
the  mercurial  column  forming  the  requisite  compensation. 
Mr.  Baily's  more  accurate  calculation  gives  6'31  inches. 

A  mercurial  compensation  pendulum  may  be  formed,  hav- 
ing a  cylinder  of  steel  or  iron,  with  its  top  constructed  in  the 
same  manner  as  the  top  of  the  stirrup,  so  as  to  receive  the 
screw  of  the  rod.  To  find  the  length  of  the  mercurial 
column  necessary  in  a  pendulum  of  this  description  (that  is, 
with  a  cylinder  made  of  steel),  we  must  double  the  linear 
expansion  of  steel,  and  take  it  from  the  absolute  expansion 
of  mercury,  to  obtain  the  relative  vertical  expansion  of  the  mer- 
cury. This  will  be  -00010010—  -00001272  =  -00008738; 

,.  ,     f.  -0000063596  r\~/c\*tt\ 

and,  proceeding  as  before,  we  have    .00008738   —  '0/279. 

Let  the  length  of  the  steel  rod  be,  as  before,  42  inches. 
Multiplying  this  by  -07279,  we  have  3-057,  which  being 
doubled,  and  one  tenth  of  the  product  added,  we  obtain 
6-72  inches  for  the  length  of  the  compensating  mercurial 
column  ;  which  Mr.  Baily  states  to  be  6-59. 

A  mercurial  compensation  pendulum,  having  a  rod  of  glass, 
has  been  employed  by  the  writer  of  this  article,  who  has  had 
reason  to  think  well  of  its  performance.  Its  cheapness  and 
simplicity  much  recommend  it.  It  is  merely  a  cylinder  of 
glass  of  about  7  inches  in  depth,  and  2£  inches  diameter, 
terminated  by  a  long  neck,  which  forms  the  rod  of  the  pen- 
dulum, the  whole  blown  in  one  piece.  A  cap  of  brass  is 
clamped  by  means  of  screws  to  the  top  of  the  rod,  and  to 
this  the  pendulum  spring  is  pinned. 

We  have  unquestionable    authority   for   saying,   that  the 


CHAP.  XXI.       COMPENSATION  PENDULUM.  281 

mercurial  pendulum  of  the  usual  construction,  that  is,  with 
a  steel  rod  and  glass  cylinder,  is  not  affected  by  a  change  of 
temperature  simultaneously  in  all  its  parts.  Now,  the  pen- 
dulum of  which  we  are  treating  being  formed  throughout  of 
the  same  material  in  a  single  piece,  and  in  every  part  of  the 
same  thickness,  it  is  presumed  it  cannot  expand  in  a  linear 
direction,  until  the  temperature  has  penetrated  to  the  whole 
interior  surface  of  the  glass,  when  it  is  rapidly  diffused 
through  the  mass  of  mercury.  M.  Biot  mentions  that  a 
pendulum  of  this  kind  was  formerly  used  in  France,  and 
expresses  his  surprise  that  it  was  no  longer  employed,  as 
he  had  heard  it  very  highly  spoken  of.  The  writer  of  this 
article  has  also  used  a  pendulum  with  a  glass  rod,  which 
differs  from  that  we  have  just  mentioned,  in  having  the  lower 
end  of  the  rod  firmly  fixed  in  a  socket  attached  to  the  centre 
of  a  circular  iron  plate,  on  the  circumference  of  which  a 
screw  is  cut,  which  fits  into  a  collar  of  iron,  supporting  the 
cylinder  (to  which^t  is  cemented)  by. means  of  a  circular  lip 
This  arrangement,  though  perhaps  less  perfect  than  that 
we  have  just  described,  the  pendulum  not  being  in  one  piece, 
has  the  advantage  of  allowing  a  circular  plate  of  glass  to  be 
placed  upon  the  surface  of  the  mercury,  as  practised  by  Mr. 
Browne.  To  determine  the  length  of  a  column  of  mercury 
for  a  glass  pendulum,  let  us  suppose  the  glass,  including  the 
cylinder,  to  be  41  inches  in  length.  Multiplying  this  by 
•0529,  the  number  taken  from  Table  II.  for  a  glass  rod  and 
mercury  in  a  glass  cylinder,  we  have  2*17  inches  for  the  un- 
corrected  length  of  mercury,  which  compensates  41  inches 
of  glass.  Suppose  the  steel  spring  to  be  one  inch  and  a  half 
long :  multiplying  this  by  -0703,  the  appropriate  decimaj 
taken  from  Table  II.,  we  haveO'l,  the  length  of  mercury 
-me  to  the  steel,  making  with  the  former  2*27  inches,  which 
being  doubled,  and  the  product  increased  by  its  one  tenth 
part,  we  obtain  five  inches  for  the  length  of  the  required 
column  of  mercury. 

Compensation  Pendulum  of  Wood  and  Lead,  on  the  Princi- 
ple of  the  Mercurial  Pendulum. 

If  by  any  contrivance  wood  could  be  rendered  impervious 
to  moisture,  it  would  afford  one  of  the  most  convenient  sub- 
stances known  for  a  compensation  pendulum.  It  does  not 
appear  that  sufficient  experiments  have  been  made  upon  this 

24* 


282  THE    ELEMENTS    OF    MECHANICS.  CHAP.    XXI. 

subject  to  decide  the  question.  Mr.  Browne  of  Portland 
Place,  who  has  devoted  much  of  his  time  and  attention  to  the 
most  delicate  inquiries  of  this  kind,  has,  we  believe,  found 
that  if  a  teak  rod  is  well  gilded,  it  will  not  afterwards  be 
affected  by  moisture.  At  all  events,  it  makes  a  far  superior 
pendulum,  when  thus  prepared,  to  what  it  does  when  such 
preparation  is  omitted. 

Mr.  Daily,  in  the  paper  we  have  before  alluded  to,  pro- 
poses an  economical  pendulum  to  be  constructed  by  means 
of  a  leaden  cylinder  and  a  deal.  rod.  He  prefers  lead  to 
zinc,  on  account  of  its  inferior  price,  and  the  ease  with 
which  it  may  be  formed  into  the  required  shape  ;  and  as 
there  is  no  considerable  difference  in  their  rates  of  expansion, 
it  is  equally  applicable  to  the  purpose. 

Let  the  length  of  the  deal  rod  be  taken  at  46  inches. 
Then,  to  find  the  length  of  the  cylinder  of  lead  to  compen- 
sate this,  we  have,  in  Table  II.,  -1427  for  such  a  pendulum ; 
which,  being  multiplied  by  40,  the  product  doubled,  and  one 
tenth  of  the  result  added  to  it,  gives  14-44  inches  for  the 
length  of  the  leaden  cylinder.  Mr.  Baily's  compensation 
gives  14-3  inches. 

The  rod  is  recommended  to  be  made  of  about  three 
eighths  of  an  inch  in  diameter  :  the  leaden  cylinder  is  to  be 
cast  with  a  hole  through  its  centre,  which  will  admit  with 
perfect  freedom  the  cylindrical  end  of  the  rod.  The  cylinder 
is  supported  upon  a  nut,  which  screws  on  the  end  of  the  rod 
in  the  usual  manner.  This  pendulum  is  represented  at 
fig-  224. 

Mr.  Baily  proposes  that  the  pendulum  should  be  adjusted 
nearly  to  the  given  rate  by  means  of  the  screw  at  the  bottom, 
and  that  the  final  adjustment  be  made  by  means  of  a  slider 
moving  along  the  rod.  Indeed,  this  is  a  means  of  adjust- 
ment which  we  would  recommend  to  be  employed  in  every 
pendulum. 

Smeaton's  Pendulum. 

We  shall  conclude  our  account  of  compensation  pendulums 
with  a  description  of  that  invented  by  Mr.  Smeaton.  The 
compensation  for  temperature  in  this  pendulum  is  effected 
by  combining  the  two  modes,  which  have  been  so  fully  de- 
scribed in  the  preceding  part  of  this  article. 

The  pendulum  rod  is  of  solid  glass,  and  is  furnished  with  a 
steel  §crew  and  nut  at  the  bottom  in  the  usual  manner.  Upon 


CHAP.  xxi.  SMEATON'S  PENDULUM.  283 

the  glass  rod  a  hollow  cylinder  of  zinc,  about  the  eighth  of 
an  inch  thick,  and  about  12  inches  long,  passes  freely,  and 
rests  upon  the  nut  at  the  bottom  of  the  pendulum  rod. 

Over  the  zinc  cylinder  passes  a  tube  made  of  sheet-iron. 
The  edge  of  this  tube  at  the  top  is  turned  inwards,  and  is 
notched  so  as  to  allow  of  this  being  effected.  A  flanche  is 
thus  formed,  by  which  the  iron  tube  is  supported,  upon  the 
zinc  cylinder.  The  lower  edge  of  the  iron  tube  is  turned 
outwards,  so  as  to  form  a  base  destined  to  support  a  leaden 
cylinder,  which  we  are  about  to  describe. 

A  cylinder  of  lead,  rather  more  than  12  inches  long,  is 
cast  with  a  hole  through  its  axis,  of  such  a  diameter  as  to 
allow  of  its  sliding  freely,  but  without  shake,  upon  the  iron 
tube  over  which  it  passes,  and  by  the  lower  extremity  of 
which  it  is  supported. 

Now  the  zinc,  resting  upon  the  nut,  and  expanding  up- 
wards, will  raise  the  whole  of  the  remaining  part  of  the 
compensation.  This  expansion  upwards  will  be  slightly 
counteracted  by  the  lesser  expansion  downwards  of  the  iron 
tube,  which  carries  with  it  the  leaden  cylinder.  The  cylin- 
der of  lead  now  acts  upon  the  principle  of  the  mercurial 
pendulum,  and,  expanding  upwards,  contributes  that  which 
was  wanting  to  restore  the  centre  of  oscillation  to  its  proper 
distance  from  the  point  of  suspension. 

This  pendulum,  we  have  been  informed,  does  well  in  prac- 
tice, and  we  are  not  aware  that  a*y  description  of  it  has  been 
before  published. 

The  method  of  calculating  the  length  of  the  tubes  required 
to  form  the  compensation,  is  very  simple ;  nothing  more  is 
necessary  than  to  find  the  length  of  zinc,  the  expansion  of 
which  is  equal  to  that  of  the  pendulum  rod. 

Let  the  pendulum  rod  be  composed  of  43  inches  of  glass, 
the  spring  being  an  inch  and  a  half  long,  and  the  screw  be- 
tween the  end  of  the  glass  rod  and  the  nut  half  an  inch, 
making  in  the  whole  two  inches  of  steel  and  43  inches  of 
glass. 

Now,  to  find  the  length  of  zinc  that  will  compensate  the 
glass,  we  have,  from  Table  II.,  for  glass  and  zinc  -2773, 
which  multiplied  by  43,  gives  11*92  inches.  In  like  manner 
we  obtain  as  a  compensation  for  two  inches  of  steel  0*74  of 
zinc,  which,  added  to  11-92,  gives  12-66  inches  for  the  total 
length  of  the  zinc  cylinder. 

Now,  if  the  iron  tube  and  the  lead  cylinder  be  each  made 


284  THE    ELEMENTS    OF    MECHANICS.  CHAP.    XXI. 

of  the  same  length  as  the  zinc,  and   arranged  as  we  have 
described,  the  compensation  will  be  perfect. 

To  prove  this,  find,  by  means  of  the  expansions  given  in 
Table  I.,  the  actual  expansion  of  each  of  the  substances  em- 
ployed in  the  pendulum,  and  we  shall  have  the  following 
results : — 

The  expansion  of  12*66  inches  of  zinc  expanding 

upwards  is -0002186 

Deduct  that  of  12-66  inches  of  iron  expanding 

downwards  .  .  -0000869 


Remaining  effect  of  expansion  upwards,  referred 

to  the  lower  extremity  of  the  iron  tube -0001317 

Now,  for  the  lead. — On  the  principle  of  the  mer- 
curial compensation,  subtract  one  tenth  part  of 
the  length  of  the  cylinder,  and  take  half  the 
remainder,  and  we  shall  have  six  inches  of  lead, 
the  expansion  of  which  upwards  is -0000955 


Total  expansion  of  the  compensation  upwards    .  .  -0002272 

To  find  the  expansion  of  the  rod,  we  have  the  ex- 
pansion of  43  inches  of  glass -0002059 

Of  two  inches  of  steel  .  -0000127 


Total  expansion  of  the  pendulum  rod -0002186 

Agreeing  near  enough  with  that  of  the  compensation  before 
found. 

As  we  conceive  we  have  been  sufficiently  explicit  in  our 
description  of  this  pendulum,  in  the  construction  of  which 
no  difficulty  presents  itself,  we  think  an  engraved  representa- 
tion of  it  would  be  superfluous. 

We  have  hitherto  treated  only  of  compensations  for  tem- 
perature ;  but  there  is  another  kind  of  error,  which  has  been 
sometimes  insisted  upon,  arising  from  a  variation  in  the 
density  of  the  atmosphere.  If  the  density  of  the  atmos- 
phere be  increased,  the  pendulum  will  experience  a  greater 
resistance,  the  arc  of  vibration  will  in  consequence  be  dimin- 
ished, and  the  pendulum  will  vibrate  faster.  This,  however, 
is  in  some  measure  counteracted  by  the  increased  buoyancy 
of  the  atmosphere,  which,  acting  in  opposition  to  gravity, 


CHAP.    XXI.  PENDULUMS.  285 

occasions  the  pendulum  to  vibrate  slower.  If  the  one  effect 
exactly  equalled  the  other,  it  is  evident  no  error  would  arise  ; 
and  in  a  paper  by  Mr.  Davies  Gilbert,  President  of  the  Royal 
Society  of  London,  published  in  the  Quarterly  Journal  for 
1826,  he  has  proved  that,  by  a  happy  chance,  the  arc  in 
which  pendulums  of  clocks  are  usually  made  to  vibrate,  is  the 
arc  at  which  this  compensation  of  error  takes  place.  This 
arc,  for  a  pendulum  having  a  brass  bob,  is  1°  56'  30"  on  each 
side  of  the  perpendicular ;  and  for  a  mercurial  pendulum, 
1°  31'  44",  or  about  one  degree  and  a  half. 

It  is  well  known  that,  if  a  pendulum  vibrates  in  a  circular 
arc,  the  times  of  vibration  will  vary  nearly  as  the  squares  of 
the  arcs;  but  if  the  pendulum  could  be  made  to  vibrate  in  a 
cycloid,  the  time  of  its  vibration  in  arcs  of  different  extent 
would  then  remain  the  same.  Huygens  and  others,  therefore, 
endeavored  to  effect  this  by  placing  the  spring  of  the  pendu-" 
lum  between  checks  of  a  cycloidal  form. 

When  escapements  are  employed  which  do  not  insure  an 
unvarying  impulse  to  the  pendulum,  the  force  may  be  un- 
equally transmitted  through  the  train  of  the  clock  in  conse- 
quence of  unavoidable  imperfections  of  workmanship,  and 
the  arc  of  vibration  may  suffer  some  increase  or  diminution 
from  this  cause.  To  discover  a  remedy  for  this  is  certainly 
desirable. 

The  writer  of  this  article  some  years  ago  imagined  a 
mode,  which  he  believes  has  also  been  suggested  by  others, 
by  which  he  conceived  a  pendulum  might  be  made  to  de- 
scribe an  arc  approaching  in  form  to  that  of  a  cycloid.  The 
pendulum  spring  was  of  a  triangular  form,  and  the  point  or 
vertex  was  pinned  into  the  top  of  the  pendulum  rod,  the  base 
of  the  triangle  forming  the  axis  of  suspension.  Now,  it  is 
evident  that  when  the  pendulum  is  in  motion,  the  spring  will 
resist  bending  at  the  axis  of  suspension,  with  a  force  in  some 
sort  proportionate  to  the  base  of  the  triangle. 

Suppose  the  pendulum  to  have  arrived  at  the  extent  of  its 
vibrations  ;  the  spring  will  present  a  curved  appearance  ;  and 
if  the  distance  from  the  point  of  suspension  to  the  centre  of 
oscillation  be  then  measured,  it  will  evidently,  in  consequence 
of  the  curvature  of  the  spring,  be  shorter  than  the  distance 
from  the  point  of  suspension  to  the  centre  of  oscillation, 
measured  when  the  pendulum  is  in  a  perpendicular  position, 
and  consequently  when  the  spring  is  perfectly  straight. 

The  base  of  the  triangle  may  be  diminished,  or  the  spring 


286  THE  ELEMENTS  OF  MECHANICS.  CHAP.  XXI. 

be  made  thinner  ;  either  of  which  will  lessen  its  effect.  We 
cannot  say  how  this  plan  might  answer  upon  further  trial, 
as  sufficient  experiments  were  not  made  at  the  time  to  au- 
thorize a  decisive  conclusion. 

We  have  thus  completed  our  account  of  compensation 
pendulums  ;  but  before  we  conclude,  it  may  not  be  unaccep- 
table if  we  offer  a  few  remarks  on  some  points  which  may 
be  found  of  practical  utility. 

The  cock  of  the  pendulum  should  be  firmly  fixed  either 
to  the  wall  or  to  the  case  of  the  clock,  and  not  to  the  clock 
itself,  as  is  sometimes  done,  and  which  has  occasioned  much 
irregularity  in  its  rate,  from  the  motion  communicated  to  the 
point  of  suspension.  We  prefer  a  bracket  or  shelf  of  cast 
iron  or  brass,  upon  which  the  clock  may  be  fixed,  and  the 
cock  carrying  the  pendulum  attached  to  its  perpendicular 
back.  This  bracket  may  either  be  screwed  to  the  back  of 
the  clock-case,  or,  which  is  the  better  mode,  securely  fixed 
to  the  wall ;  and  if  the  latter  be  adopted,  the  whole  may  be 
defended  from  the  atmosphere,  or  from  dust,  by  the  clock-case, 
which  thus  has  no  connection  either  with  the  clock  or  with 
the  pendulum. 

The  point  of  suspension  should  be  distinctly  defined  and 
immovable.  This  may  be  readily  effected,  after  the  pendu- 
lum shall  have  taken  the  direction  of  gravity,  by  means  of  a 
strong  screw  entering  the  cock  (which  should  be  very  stout) 
on  one  side,  and  pressing  a  flat  piece  of  brass  into  firm  con- 
tact with  the  spring. 

The  impulse  should  be  given  in  that  plane  of  the  rod 
which  coincides  with  the  plane  of  vibration  passing  through 
the  axis  of  the  rod.  If  the  impulse  be  given  at  any  point 
either  before  or  behind  this  plane,  the  probable  result  will  be 
a  tremulous,  unsteady  motion  of  the  pendulum. 

A  few  rough  trials,  and  moving  the  weight,  will  bring  the 
pendulum  near  its  intended  time  of  vibration,  which  should 
be  left  a  little  too  slow  ;  when  the  bob  should  be  firmly  fixed 
to  the  rod,  if  the  form  of  the  pendulum  will  admit  of  it,  by 
a  pin  or  screw  passing  through  its  centre. 

The  more  delicate  adjustment  may  be  completed  by  shift- 
ing the  place  of  the  slider  with  which  the  pendulum  is  sup- 
posed to  be  furnished  on  the  rod. 

Mr.  Browne  (of  whom  we  have  before  spoken)  practises 
the  following  very  delicate  mode  of  adjustment  for  rate,  which 
will  be  found  extremely  convenieat,  as  it  is  not  necessary  to 


CHAP.    XXI.  PENDULUMS.  287 

stop  the  pendulum  in  order  to  make  the  required  alteration. 
Having  ascertained,  by  experiment,  the  effect  produced  on 
the  rate  of  the  clock,  by  placing  a  weight  upon  the  bob  equal 
to  a  given  number  of  grains,  he  prepares  certain  smaller 
weights  of  sheet-lead,  which  are  turned  up  at  the  corners, 
that  they  may  be  conveniently  laid  hold  of  by  a  pair  of  for- 
ceps, and  the  effect  of  these  small  weights  on  the  rate  of  the 
clock  will  be,  of  course,  known  by  proportion.  The  rate 
being  supposed  to  be  in  defect,  the  weights  necessary  to  cor- 
rect this  may  be  deposited,  without  difficulty,  upon  the  bob 
of  the  pendulum,  or  upon  some  convenient  plane  surface, 
placed  in  order  to  receive  them  :  and  should  it  be  necessary 
to  remove  any  one  of  the  weights,  this  may  readily  be  done 
by  employing  a  delicate  pair  of  forceps,  without  producing 
the  slightest  disturbance  in  the  motion  of  the  pendulum. 


INDEX. 


A. 

Action  and  reaction,  29. 

Aeriform  fluids,  22. 

Animalcules,  10. 

Atmosphere,  impenetrability  of,  15. 
Compressibility  and  elasticity  of,  1C. 

Atoms,  5.    Coherence  of,  5. 

Attraction,  magnetic,  of  gravitation,  7, 
41,  53.  Molecular  or  atomic,  57.  Co- 
hesion, 58. 

Attwood,  machine  of,  76. 

Axes,  principal,  117. 

Axis,  mechanical  properties  of,  108. 

B. 

Balance,235.  Of  Bate,  242.  Use  of,  213. 
Danish,  252.  Bent-lever  of  Brady,  253. 

Bodies,  1.  Lines,  surfaces,  edges,  area, 
length  of,  3.  Figure,  volume,  shape 
of,  4.  Porosity  of,  14.  Compressibili- 
ty of,  16.  Elasticity,  dilatability  of,  16. 
Inertia  of,  23.  Rule  for  determining 
velocity  of  motion  of  two  bodies  after 
impact,  32. 

C. 

Capillary  attraction,  61. 

Capstan,  151. 

Cause  and  effect,  6. 

Cir-le  of  curvature,  84. 

Cog,  hunting,  161. 

Components.  42. 

Cord,  139. 

Cordage,  friction  and  rigidity  of,  219. 

Crank,  203. 

Crystallization,  12. 

Cycloid,  134. 

D. 

Damper,  self-acting,  197. 

Deparcieux's  compensation   pendulum, 

269. 

Diagonal,  42. 
Dynamics,  135. 
Dynamometer,  257. 


Electricity,  63. 
Eleetro  magnetism,  63. 
Equilibrium,  neutral,  instable  and 
ble,  99. 


sta- 


Figure, 4. 
Yly. wheel,  201. 


Force,  5.  Composition  and  resolution 
of,  41.  Centrifugal,  84.  Moment  of; 
leverage  of,  114.  Regulation  and  ac- 
cumulation of,  189. 

Friction,  effects  of,  82.    Laws  of,  223. 

G. 

Governor,  191. 

Gravitation,  attraction  of,  64.  Terres- 
trial, 70. 

Gravity,  centre  of,  90. 
Gyration,  radius  of,  centre  of,  115 

II . 

Hooke's  universal  joint,  212. 
Hydrophane,  porosity  of,  15. 


Impact,  33. 
Impulse,  53. 

Inclined  plane,  138,  176. 
Inclined  roads,  177. 

Inertia,  23.    Laws  of,  28.    Moment  of, 
116. 

J. 
Julien  le  Roy ,  compensation  tube  of,  268. 


Lever,  138.    Fulcrum  of;  three  kinds  of, 

141.    Equivalent,  148. 
Line  of  direction,  93. 
Liquids,  compressibility  of,  20. 
Loadstone,  57. 

M. 

Machines,  simple,  135.  Power  of,  148 
Regulation  of,  189. 

Magnet,  56. 

Magnetic  attraction,  7. 

Magnetism,  63. 

Magnitude,  3. 

Marriott's  patent  weighing  machine,  257. 

Materials,  strength  of,  229. 

Matter,  properties  of,  1.  Impenetrability 
of,  4.  Atoms  of;  molecules  of,  5. 
Divisibility  of,  7.  Examples  of  the 
subtilty  of,  10.  Limit  to  the  divisibility 
of,  11.  Porosity  of;  density  of,  14. 
Compressibility  of,  16.  Elasticity  and 
dilatability  of,  16.  Impenetrability  of, 
19.  Inertia  of,  23. 

Mechanical  science,  foundation  of,  14. 


INDEX. 


Metronomes,  principles  of,  130. 

Molecules,  5. 

Motion,  laws  of,  38.  Uniformly  accel- 
erated, 73.  Table  illustrative  of,  75. 
Retarded,  78.  Of  bodies  on  inclined 
planes  and  curves,  79.  Rotary  and 
progressive,  107.  Mechanical  con- 
trivances for  the  modification  of,  206. 
Continued  rectilinear;  reciprocatory 
rectilinear ;  continued  circular  ;  recip- 
rocating^circular,  207. 
N. 

Newton,  method  of,  for  determining  the 
thickness  of  transparent  substances,  9. 
Laws  of  motion  of,  38. 

O. 

Oscillation,  109.  Of  the  pendulum,  123. 
Centre  of,  128. 

P. 

Parallelogram.  42. 

Particle,  5. 

Pendulum,  oscillation  or  vibration  of, 
123.  Isochronism  of,  125.  Centre  of 
oscillation  of,  124.  Of  Tronghton,  239. 
Compensation,  259.  Of  Harrison,  264. 
Tubular,  of  Troughton,264.  Of  Ben- 
zenberg,  266.  Ward's  compensation, 

268.  Captain  Rater's  compensation, 

269.  Reid's  compensation, 271.    Elli- 
cott's,  272.     Steel  and  brass  compensa- 
tion, 273.     Mercurial,  275.     Graham's 
mercurial,  277.     Wood  and   load, 281. 
Smeaton's,282. 

Percussion,  109.    Centre  of,  192. 

Planes  of  cleavage,  13. 

Porosity,  14. 

Power,  136. 

Properties,  1. 

Projectiles,  curvilinear  path  of,  68. 

Pulley,  139.    Tackle  ;  fixed,  167.    Single 

movable,     168.        Called    a    runner; 

Spanish  bartons,  173. 

R. 

Rail-roads,  177. 
Regulating  damper,  197. 


Regulators,  191. 
Repulsion,  7.     Molecular,  61. 
Resultant,  42. 
Rose-engine,  211. 

S. 

Salter,  spring  balance  of,  256. 

Screw,  176.  Concave,  183.  Microme- 
ter, 188. 

Shape,  5. 

Spring,  256. 

Statics,  135. 

Steelyard,  248.  C.  Paul's,  249.  Chi- 
nese, 252. 

Syphon,  capillary,  61. 

T. 

Table,  whirling,  85. 
Tachometer,  197. 
Tread-mill,  151. 

V. 

Velocity,  angular,  84. 

Vibration,  109.    Of  the  pendulum,  123. 

Centre  of,  128. 
Volume,  4,  14. 

W. 

Watch,  main-spring  of;  balance-wheel 
of,  164. 

Water  regulator,  193. 

Wedge,  176.    Use  of,  180. 

Weight,  136. 

Weighing  machines,  234.  For  turnpike- 
roads,  254.  By  means  of  a  spring,  255. 

Wheels,  spur,  crown,  bevelled,  159.  Es- 
capement, 164. 

Wheel  and  axle,  149 

Wheel- work,  149. 

Winch,  151. 

Windlass,  151. 

Wollaston's  wire,  9. 

Z. 

Zureda,  apparatus  of;  Leupold'd  appli- 
cation of,  211. 


THE  END. 


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RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 


This  book  is  due  on  the  last 

the  date 


i  1972  70 


LD21A-60m-8,'70 
(N8837slO)476 — A-32 


General  Library 

University  of  Californis 

Berkeley 


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